Funny enough my partner actually did this project when she was in high school, so I have some suggestions. A good way of testing might be to get white tiles (like from Home Depot) and then stain them with common foods. Wine or coffee might be good examples. Then, you can brush each tile with a different toothpaste and see which works best at cleaning it. Generally, two things whiten teeth in toothpaste. One is rough sand-like additives that scrape away at stains. The second is by including chemicals that break down the stains by reacting with them. These chemicals can vary, but they are usually called oxidizing agents. You should be able to find them on the back of the toothpaste you experiment with.

The chemical in toothpaste and other tooth-whitening products is usually hydrogen peroxide. Hydrogen peroxide is a bleaching agent, and is mostly safe when it's dilute (although you still don't want more than a tiny bit getting swallowed). Some toothpastes combine hydrogen peroxide and baking soda.

You could compare toothpastes that contain hydrogen peroxide with those that don't (probably some of the 'natural' brands), and you can also try using just plain peroxide and just plain baking soda/water to see how the individual ingredients contribute to the whitening.

You'll probably want to compare the treatments on something that's not your mouth for a controlled scientific study. :) As an idea, you can brush a set of coffee-stained mugs and compare before and after photos to see which treatment removed the most stains. Good luck with the project!