I read that you can use Beets to make a natural Red food coloring, how would I go about doing this?
Anything that's hard to get off of your counter can make a good food coloring. Regarding natural sources, remember that you run the risk of also adding additional flavors so take that into consideration. Among other companies, Annie's makes all natural foods including colorful fruit snacks. What they use for the coloring in these snacks are: turmeric, black carrot juice concentrate, and annatto. (I've cooked with turmeric and annatto before and they DEFINITELY stain.) Aside from Google searches, you could probably find practical natural food dyes by looking in the ingredient list from other all-natural foods.
Back to your specific question about beets, this is from eHow: