A couple of people have suggested starting with going fishing and I think thats a good idea. But theres an even easier and more available place to start that you mentioned in your post. Go to the grocery and buy a whole chicken and practice processing it. Not only is it a good way to build your ability to deal with butchering an animal. Its also a good way to learn about the anatomy of small animals and how to process them for cooking. After that, a rabbit is pretty much the same except you have to peel the skin off first and their leaner so cooking them with some bacon fat is a definite plus.

I also highly recommend the book The Omnivore's Dilemma its mostly about where our industrial food supply comes from and how its made, but theres also an interesting piece on small scale butchering and also on hunting. A very good read and a real eye opener on where our food comes from.