Is there any way you can add horse care to your lessons? Riding, grooming and leading are purely basics and with just those under your belt, you really don't have the knowledge to own and take care of your own horse. See if you can pay your instructor to take you further in your knowledge with the goal of owning your own. If you have a good one, they will tell you when you are ready and probably help you find the perfect horse.
Working with/or for a horse owner would give you a realistic picture of what it's like and the sacrifices that must be made. Do you have anyone at the stable where you ride that you can work for and learn from? Are there horses for lease where you can contribute time to their care?
Even for those of us with decades of experience, owning a horse can be a hardship-time, money and pure work. You really don't want to actually hurt or neglect a horse you buy just from pure lack of knowledge. Horse ownership costs have been rising constantly-mine are costing me $2000 more a year than I was paying 2 years ago and I have not changed anything! These animals are incredibly expensive and that's without the unexpected vet bill!
There are some great, calm, sensible TBs out there and OTTBs are extremely cheap but in general, they are the worst breed a beginner/novice can buy.
I love them myself but in general they are hard keepers and most are extremely high strung. If they come straight off the track, they will require transition training to become safe riding horses.
Gather your knowledge, ask questions, work with them with the goal in mind of buying one in the future. With what you know now, you'd do a horse more harm than good.