I think leasing would be a better first option; however, it mostly depends on what you want to do, and what you can handle. I personally recommend Quarter Horses for a starting horse – they're incredibly versatile, and if you want to change your focus, you can. Any horse is fine though, just make sure to look into their temperament, and so on, and check for previous injuries.
I think that you should LEASE a quarter horse that's older than 10. If you are to buy a horse, make sure that you are at an experienced barn that will make sure your horse is ok and I'll teach you everything you need to know! Good luck!!
Try leasing out a horse and do your research on horses and learn that horses each have there own personality and it's not only the breed that makes it a horse for a beginner. If you have the money and think your ready for a horse. Take more lessons and put that towards knowledge because once you start taking more lessons and gain more knowledge you might want to consider
I don't think you're ready for a horse yet. If you have just started getting lessons you shouldn't lease/buy a horse. You need more knowledge about horses. Having a horse is much more than just riding it and feeding it. Maybe ride a few more years, get to know horses better, and shadow some vets first. But for a beginner, you want a horse that is calm and won't spook at stuff. Preferably an older horse is better than a younger one. They should be able to teach the rider. Horses are all different so you can't judge them by the breed.
What I would do would be to ask your trainer if she knows of anyone who is leasing a horse in your area. It's like owning a horse without the commitment so you can see what ownership is like without doing anything too drastic