If you feel you may be biting off more than you can chew, I'd start with a lease for awhile and see how things go. I started riding when I was 8, because my best friend owned horses. When I was 14, my dad leased a horse for me. I leased it for a few years, showed, took lessons twice a week, was very much into it. At 18 I went to college, and that ended my horse adventure until just his year at age 38! My dad was very smart in knowing that leasing was the best choice for our family, as when I went to college we walked away from the lease and he wasnt left with a horse to sell.
I currently lease, take lessons and learn everything I can while I prepare to finally purchase a horse in the next year or so. It has given me the opportunity to absolutely be sure I want the responsibility of ownership (which I do). Once again, its the right thing for my family, as the purchase of a horse is going to be a huge financial commitment for me, my husband and my kids.
Not saying leasing is the right thing for everyone, but it does allow a child to "prove" they are committed to caring for and remaining involved long term with the animal before jumping into ownership.