One of the easiest exercises I know, and one that will certainly wake up your eye to see the Rule of Thirds, involves the use of blue-, red-, green-, and yellow-colored construction paper. Start with a sheet of blue paper and fill up your frame with the blue. Feel free to crank up the ISO and shoot handheld near a window, or do this outside, weather permitting.
Now take the green paper and cut out four strips, each 1/2-inch wide and 11 inches long. Place just one of those strips so it runs through the middle of the red paper and take a shot. Now take a second strip and place one across the top third of the paper and the other across the bottom third. Take another shot. Finally, leaving those two strips in place, get two more strips and place them vertically along the right third and left third (as if you were creating a tic-tac-toe board). Take a fourth shot. You just photographed the Suggestion of Thirds grid.
Now get the yellow paper and cut out a circle roughly 2 inches in diameter. Place the circle on one of the points where your vertical and horizontal lines cross and take a picture. Repeat for each point where the lines cross. For your final image, remove the strips and place just one green strip so it runs through the middle of the frame. Then place the yellow circle right in the middle so it sits atop the strip and take that shot. Now show your family or a friend the eight shots you just took and ask which arrangements are the least appealing. The odds are good that they will not like the first and last images. This has much to do with our innate sense of order and tension. In the first and last shots, you have only order: an even 50/50 division of the paper.
