There is so much more to owning a horse than just knowing how to ride. I second the suggestion to try a part lease first. In addition, if your instructor is open to it, ask for lessons in horse care and handling in addition to the riding. In the meantime, I would be researching everything I possibly could.
You'll need at least a basic understanding of horse nutrition, the knowledge of the warning signs of common ailments such as colic, how to treat minor wounds, how to know when the wound isn't minor and a vet needs to be called. You'll need to be confident in handling a horse on the ground, as even the most good-natured seeming horse may try to take advantage of you if you act timid and let him get away with things.
Find a good instructor who is happy to teach you the "other" things you'll need to know, find a good part lease on a well trained horse (once you've been riding with your instructor for awhile, they should be able to help you find a suitable mount). Figure out what discipline(s) you want to pursue so that when it does come time to get your own horse, you'll know what type of horse you want to look at. Ask tons of questions, read tons of articles and books. Read forums such as this one (taking things with a grain of salt of course or doing independent research).
Lots of people jump into horse ownership with no or very little experience and do fine, others go on to regret their lack of knowledge. They may get hurt or their horse might. Granted that still can happen when you have more experience, still I wouldn't recommend committing to something as large and strong and at the same time fragile as a horse without some serious hours spent in the saddle and around horses.