Afraid I can't help all that much, my kids have just started in 4-H and I was never in it myself. I do know that you start out by finding a group near you to join that focuses on horses. Evidently, there are various "types" of 4-H groups you can be in, some focus on horses, some on arts and crafts, some on cattle, etc.

If you call around to your local feed stores, vet clinic, etc. someone should be able to send you in the right direction! Good luck, from what I've seen and heard its a very good program to be in.