If you're leaning towards more eventing and dressage, Pony Club would probably be a better fit for you. Both 4-H and Pony Club are good organizations for kids who ride. You can't go too wrong with either one, but I prefer Pony Club. I've seen it from many different angles over a 30 year period! I joined when I was 14 and had been riding for 8 years (and also had a 14.2hh'er), and stayed in until I hit the top rating and turned 21 (the old age limit). Then, 4 years ago, I signed up my then 7 year old beginner son, so I got to experience it as a parent, and with a young beginner kid.
Within the national framework, each pony club does things a little differently. You'll have to ask questions of your local club or clubs if you have more than one to pick from. My son's club has lessons twice a week, year-round, plus unmounted meetings, fun stuff, etc. We get invited to fox hunt, and have many special opportunities. Kids compete at rallies for Quiz, Dressage, Eventing, Show Jumping, Games, Polocrosse, and Tetrathlon (riding, running, shooting & swimming). You are never required to compete, just like you're not required to rate up to new ratings until you're ready. Go to the pony club website (www.ponyclub.org) and also check out the websites of your region and local clubs. Contact the clubs you're interested in and ask as many questions as you have. Ask if you can come observe one of their activities.
Now is a good time to join because you can join as of January 1st, then be there to attend Quiz practices and make some friends in the club. If you wait until spring, you might miss the Quiz rally, and you'll pay the same annual dues as you would on Jan. 1.