It took some time, but Melburnians have embraced Federation Sq, accepting it as the congregation place it was meant to be – somewhere to celebrate, protest, watch major sporting events or just hang out. Occupying a prominent city block, Fed Square is far from square: its undulating and patterned forecourt is paved with 460,000 hand-laid cobblestones from the Kimberley region, with sight lines to Melbourne's iconic landmarks; while its buildings are clad in a fractal-patterned reptilian skin.

At the square’s street junction is the subterranean Melbourne Visitor Centre. Highly recommended free tours of Fed Square depart Monday to Saturday at 11am; spaces are limited, so get here 10 to 15 minutes early. The square has free wi-fi, and there's free daily tai chi from 7.30am and meditation at 12.30pm on Tuesday.

Hidden away in the basement of Federation Sq, the Ian Potter Centre is the second half of the National Gallery of Victoria (NGV), set up to showcase the gallery's impressive collection of Australian works. Set over three levels, it's a mix of permanent (free) and temporary (ticketed) exhibitions, comprising paintings, decorative arts, photography, prints, sculpture and fashion. There's also a great museum gift shop. Free tours are conducted daily at 11am, noon, 1pm and 2pm.

The permanent Aboriginal exhibition on the ground floor is stunning, and seeks to challenge ideas of the ‘authentic’. There are some particularly fine examples of Papunya painting and interesting use of mediums from bark and didgeridoos to contemporary sculpture and dot paintings on canvas.

Managing to educate, enthral and entertain in equal parts, ACMI is a visual feast that pays homage to Australian cinema and TV, offering a perhaps unrivalled insight into the modern-day Australian psyche. Its floating screens don't discriminate against age, with TV shows, games and movies on call, making it a great place to waste a day watching TV and not feel guilty about it. Free tours are conducted daily at 11am and 2.30pm.

The three-terraced Birrarung Marr is a welcome addition to Melbourne’s patchwork of parks and gardens, featuring grassy knolls, river promenades, a thoughtful planting of indigenous flora and great views of the city and the river. There's also a scenic walking route to the Melbourne Cricket Ground via the ‘talking’ William Barak Bridge – listen out for songs, words and sounds representing Melbourne’s cultural diversity as you walk.

Melbourne's most celebrated lane for street art, Hosier Lane's cobbled length draws camera-wielding crowds snapping edgy graffiti, stencils and art installations. Subject matter runs mostly to the political and counter cultural, spiced with irreverent humour. Pieces change almost daily (not even a Banksy is safe here). Be sure to see Rutledge Lane (which horseshoes around Hosier), too.

If ever there was a true symbol of the city, Flinders Street Station would have to be it. Built in 1854, it was Melbourne’s first railway station, and you’d be hard pressed to find a Melburnian who hasn’t uttered the phrase ‘Meet me under the clocks’ at one time or another (the popular rendezvous spot is located at the station's front entrance). Stretching along the Yarra, it's a beautiful neoclassical building topped with a striking octagonal dome.

Opposite Federation Sq stands the magnificent Anglican St Paul’s Cathedral. Services were celebrated on this site from the city’s first days. Built between 1880 and 1891, the present church is the work of distinguished ecclesiastical architect William Butterfield (a case of architecture by proxy, as he did not condescend to visit Melbourne, instead sending drawings from England). It features ornate stained-glass windows, made between 1887 and 1890, and holds excellent music programs.

Across the street from Flinders Street Station is a pub known less for its beer (served up since 1861) than its iconic nude painting of the teenaged Chloe by Jules Joseph Lefebvre. Chloe's yearning gaze, cast over her shoulder and out of the frame, was a hit at the Paris Salon of 1875.

The fine neoclassical architecture of the Old Treasury (c 1862), designed by JJ Clarke, is a telling mix of hubris and functionality. The basement vaults were built to house the millions of pounds worth of loot that came from the Victorian goldfields and now feature multimedia displays telling gold-rush stories. Also downstairs is the charmingly redolent reconstruction of the 1920s caretaker’s residence, which beautifully reveals what life in Melbourne was like in the early part of last century.

The grand steps of Victoria’s parliament (c 1856) are often dotted with slow-moving, tulle-wearing brides smiling for the camera, or placard-holding protesters doing the same. Entry inside is only by tour (free), and you'll see exuberant use of ornamental plasterwork, stencilling and gilt full of gold-rush-era pride and optimism. Building began with the two main chambers: the lower house (now the legislative assembly) and the upper house (now the legislative council).

Australia’s first federal parliament sat here from 1901, before moving to Canberra in 1927. Though they’ve never been used, gun slits are visible just below the roof, and a dungeon is now the cleaners’ tearoom.

Chinese miners arrived in search of the ‘new gold mountain’ in the 1850s and settled along this section of Little Bourke St, now flanked by traditional red archways. The Chinese Museum ( MAP GOOGLE MAP ; %03-9662 2888; www.chinesemuseum.com.au; 22 Cohen Pl; adult/child $8/6; h10am-5pm) does a wonderful job of putting it into context with five floors of displays, including gold-rush artefacts, dealings under the xenophobic White Australia policy and the stunning 63m-long, 200kg Millennium Dragon that bends around the building – it needs eight people to hold up its head alone.

A big player in Melbourne's achievement of being named Unesco City of Literature in 2008, the State Library has been at the forefront of Melbourne's literary scene since it opened in 1854. With over two million books in its collection, it's a great place to browse.

Its epicentre, the octagonal La Trobe Reading Room, was completed in 1913. Its reinforced-concrete dome was, at the time, the largest of its kind in the world and natural light illuminates its ornate plasterwork and the studious Melbourne writers who come here to pen their works.

Another highlight is the collection of Ned Kelly memorabilia, including his suit of armour.

Built in 1841, this forbidding bluestone prison was in operation until 1929. It's now one of Melbourne's most popular museums, where you can tour the tiny, bleak cells. Around 135 people were hanged here, including Ned Kelly, Australia's most infamous bushranger, in 1880. One of his death masks is on display.

With over 600 traders, the Vic Market is the largest open-air market in the southern hemisphere and attracts thousands of shoppers. It's where Melburnians sniff out fresh produce among the booming cries of spruiking fishmongers and fruit-and-veg vendors. The wonderful deli hall (with art deco features) is lined with everything from soft cheeses, wines and Polish sausages to Greek dips, truffle oil and kangaroo biltong.

Originally known as Burial Hill, these gardens were the site of Melbourne's first cemetery – eight of the city’s early settlers were buried here. Today its pleasant, open lawns are popular with workers taking a lunchtime break. The gardens contain trees that are well over 100 years old, including Moreton Bay fig trees and a variety of eucalypts, including spotted gums, sugar gums and river red gums. There are plenty of possums about, but don't feed them.

Devoted to southeastern Aboriginal culture, this cultural centre displays interesting artefacts and oral history. Its gallery spaces show a variety of contemporary and traditional work, a model scar tree at the centre’s heart, and a permanent chronological display of Victorian Koorie history. Behind the scenes, significant objects are carefully preserved – replicas that can be touched by visitors are used in the displays. The centre is in the process of relocating, so check the website for details.

It also runs highly recommended tours to Flagstaff Gardens and along the Yarra, which put the areas into context with irrevocable changes to both the people and the place. Its impact on all senses evokes memories that lie beneath the modern city. Tours are mainly for school groups, but it's normally OK to tag along; call ahead to enquire.

Another reason to visit is its shop, which sells books on Aboriginal culture, CDs, crafts and bush-food supplies.

The Immigration Museum uses personal and community voices, images and memorabilia to tell the many stories of Australian immigration. Symbolically housed in the old Customs House, the restored building alone is worth the visit: the Long Room is a magnificent piece of Renaissance revival architecture.

This aquarium is home to rays, gropers and sharks, all of which cruise around a 2.2-million-litre tank, watched closely by visitors in a see-through tunnel. See the penguins in icy ‘Antarctica’ or get up close to one of Australia's largest saltwater crocs in the crocodile lair. Divers are thrown to the sharks three times a day; for between $210 and $300 you can join them. Admission tickets are cheaper online.

Head up McArthur St (the extension of Collins St) to see one of the world’s largest and finest examples of Gothic Revival architecture. Designed by William Wardell, St Patrick’s was named after the patron saint of Ireland, reflecting the local Catholic community’s main origin. Construction began in 1863 and continued until the spires were added in 1939.

Designed primarily for tourists, this free tram service travels around the city centre, passing many sights along the way with an audio commentary. It runs every 10 minutes or so.

Beyond the water-wall facade you’ll find an expansive collection set over three levels, covering international art that runs from the ancient to the contemporary. Key works include a Rembrandt, a Tiepolo and a Bonnard. You might also bump into a Monet, a Modigliani or a Bacon. It’s also home to Picasso’s Weeping Woman, which was the victim of an art heist in 1986. Free 45-minute tours run hourly from 11am to 2pm and alternate to different parts of the collection.

The Arts Centre is made up of two separate buildings: Hamer Hall (the concert hall) and the theatres building (under the spire). They're linked by a series of landscaped walkways. The George Adams Gallery and St Kilda Road Foyer Gallery are free gallery spaces with changing exhibitions. In the foyer of the theatres building, pick up a self-guided booklet for a tour of art commissioned for the building, including works by Arthur Boyd, Sidney Nolan and Jeffrey Smart.

Melbourne's tallest building, the 297m-high Eureka Tower, was built in 2006 and a wild elevator ride takes you to its 88th floor in less than 40 seconds (check out the photo on the elevator floor if there’s time). The Edge – a slightly sadistic glass cube – cantilevers you out of the building; you’ve got no choice but to look down.

ACCA is one of Australia’s most exciting and challenging contemporary galleries, showcasing a range of local and international artists. The building is, fittingly, sculptural, with a rusted exterior evoking the factories that once stood on the site, and a soaring interior designed to house often-massive installations. From Flinders St Station, walk across Princes Bridge and along St Kilda Rd. Turn right at Grant St, then left into Sturt St.

The Polly Woodside is a restored iron-hulled merchant ship (or ‘tall ship’), dating from 1885, that now rests in a pen off the Yarra River. A glimpse of the rigging makes for a tiny reminder of what the Yarra, dense with ships at anchor, would have looked like in the 19th century.

Originally erected in 2009, then disassembled due to structural problems before financial issues delayed it for several years more, the Melbourne Star Ferris wheel is finally turning. Joining the London Eye and Singapore Flyer, this giant observation wheel has glass cabins that take you up 120m for 360-degree views of the city, Port Philip Bay and further afield to Geelong and the Dandenongs. Rides last 30 minutes.

For an extra $8 you can head back for another ride at night to see the bright lights of the city.

With a capacity of 100,000 people, the 'G' is one of the world’s great sporting venues, hosting cricket in the summer and AFL footy in the winter – for many Australians it's considered hallowed ground. Make it to a game if you can (highly recommended), but otherwise you can still make a pilgrimage on non-match-day tours that take you through the stands, media and coaches’ areas, change rooms and out onto the ground (though unfortunately not beyond the boundary).

Hidden away in the bowels of the Melbourne Cricket Ground, this sports museum features five permanent exhibitions focusing on Australia’s favourite sports and celebrates historic sporting moments. Kids will love the interactive sports section where they can test their footy, cricket and netball (among other sports) skills.

The city drops away suddenly just east of Spring St, giving way to Melbourne’s beautiful backyard, the Fitzroy Gardens. The stately avenues lined with English elms, flower beds, expansive lawns, strange fountains and a creek are a short stroll from town.

The highlight is Cooks' Cottage ( GOOGLE MAP ; %03-9419 5766; adult/child/family $6/3/16.50; h9am-5pm), shipped brick by brick from Yorkshire and reconstructed in 1934 (the cottage actually belonged to the navigator’s parents). It’s decorated in mid-18th-century style, with an exhibition about Captain James Cook’s eventful, if controversial, voyages to the Southern Ocean.

The inner city melts away at this rustic riverside retreat beloved not just by children. There’s a range of frolicking farm animals that kids can help feed, as well as rambling gardens and grounds for picnicking on warm days. The farm cafe is open early and can be visited without entering the farm itself. The monthly farmers market ( GOOGLE MAP ; www.mfm.com.au; 18 St Heliers St, Abbotsford; adult/child $2/free; h8am-1pm 2nd Sat of the month), held by the river, is a local highlight, with everything from rabbits to roses to organic milk hoisted into baskets.

Foster’s beer-brewing empire runs 1½-hour tours of its Abbotsford operations, where you’ll encounter 30m-wide vats of beer and a super-fast bottling operation – and yes, samples are included in the price. Tours run Monday to Saturday; times vary so check the website. Visitors need to be aged over 18 and wearing closed-toed shoes. Bookings essential.

This not-for-profit centre has a changing schedule of photography exhibitions across a couple of galleries. Shows traverse traditional technique and the highly conceptual. There’s a particular fascination with work involving video projection, including a nightly after-hours screening in a window. Also offers photography courses.

Set in an imposing boom-style terrace, the Alcaston’s focus is on living Indigenous Australian artists. The gallery works directly with Indigenous communities and is particularly attentive to cultural sensitivities. It shows a wide range of styles, from traditional to contemporary work. There’s also a space dedicated to works on paper.

Gabrielle Pizzi, one of Australia’s most respected dealers of Indigenous art, founded this gallery in the 1980s. Her daughter Samantha continues to show contemporary city-based artists, as well as traditional artists from the communities of Balgo Hills, Papunya, Maningrida and the Tiwi Islands.

This museum provides a grand sweep of Victoria’s natural and cultural histories, with exhibitions covering everything from dinosaur fossils and giant squid specimens to a taxidermy hall, a 3D volcano and an open-air forest atrium of Victorian flora. Become immersed in the legend of champion racehorse and national hero Phar Lap. The excellent Bunjilaka, on the ground floor, presents Indigenous Australian stories and history told through objects and Aboriginal voices with state-of-the-art technology. There's also an IMAX cinema on site.

Built for the International Exhibition in 1880, and granted Unesco World Heritage status in 2004, this beautiful Victorian edifice symbolises the glory days of the Industrial Revolution, the British Empire and 19th-century Melbourne’s economic supremacy. It was the first building to fly the Australian flag, and Australia’s first parliament sat here in 1901. It now hosts everything from trade fairs to car shows. Tours of the building leave from the Melbourne Museum at 2pm.

Established in 1861, this is the oldest zoo in Australia and the third oldest in the world. Today it's one of the city’s most popular attractions. Set in spacious, prettily landscaped gardens, the zoo’s enclosures aim to simulate the animals’ natural habitats. There’s also a large collection of native animals in natural bush settings, a platypus aquarium, a fine butterfly house, fur seals, elephants, lions, tigers and plenty of reptiles.

One of the world's finest botanic gardens, the Royal Botanical Gardens are one of Melbourne’s most glorious attractions. Sprawling beside the Yarra River, the beautifully designed gardens feature a global selection of plantings and specifically endemic Australian flora. Mini-ecosystems, such as a cacti and succulents area, a herb garden and an indigenous rainforest, are set amid vast lawns. Take a book, picnic or Frisbee – but most importantly, take your time.

Beside St Kilda Rd stands the massive Shrine of Remembrance, built as a memorial to Victorians killed in WWI. It was constructed between 1928 and 1934, much of it with Depression-relief, or ‘susso’, labour. Its bombastic classical design is partly based on the Mausoleum of Halicarnassus, one of the seven ancient wonders of the world. Visible from the other end of town, planning regulations continue to restrict any building that would obstruct the view of the shrine from Swanston St as far back as Lonsdale St.

While there are palm-fringed promenades, a parkland strand and a long stretch of sand, St Kilda’s seaside appeal is more Brighton, England, than Baywatch, despite 20-odd years of glitzy development. The kiosk at the end of St Kilda Pier (an exact replica of the original, which burnt down in 2003, a year short of its centenary) is as much about the journey as the destination.

It opened in 1912 and still retains the feel of an old-style amusement park, with creepy Mr Moon’s gaping mouth swallowing you up as you enter. There’s a heritage-listed scenic railway (the world's oldest operating roller coaster) and a beautifully baroque carousel with hand-painted horses, swans and chariots, as well as the full complement of gut-churning rides.

The market’s labyrinthine interior is packed to overflowing with an eccentric collection of stalls ranging from old-school to boutique. It’s been on this site since 1864 and is a neighbourhood institution, as are its famous dim sims (sold here since 1949). There are plenty of atmospheric eateries and a lively night market on Thursdays from mid-January to early March. There’s a cooking school, too – see the website for details.

Elegant black swans give their inimitable bottoms-up salute as you jog, cycle or walk the 5km perimeter of this constructed lake. Lakeside Dr was used as an international motor-racing circuit in the 1950s, and since 1996 the revamped track has been the venue for the Australian Formula One Grand Prix each March. Also on the periphery is the Melbourne Sports & Aquatic Centre, with an Olympic-size pool and child-delighting wave machine.