There is a plant that was specifically bred to be used in short-term research. The Wisconsin Fast Plant goes from seed to seed in about 40 days.

Check it out at:
fast plants.

Bamboo is the fastest growing plant. Depending on what type of research you want to do, a lot of people have chosen Arabidopsis thaliana for gene regulation study. Good luck!

There is no easy answer to that. Some kind of annual(a plant that sprouts, grows, makes seed, and then dies) would be best, since that would do everything in a single season. Bean sprouts are good, rice is good,some sunflowers (esp. daisies) are good. You still will want weeks for your plants to sprout, though.