OP, welcome to the forum. I have to agree with what most others have said that you should discuss most of this with your instructor as they can likely tell you when you are ready for full ownership. Leasing is an excellent option as well. The most important thing you will be able to do when you finally start shopping is to find a horse who's training, temperament, and ability match your own ability and goals. For example, you really wouldn't want to look at a trail horse if you are wanting to show (even lower levels) in dressage, jumping, western pleasure, etc.
tall horses look all fine and dandy, Reality is short horses are easier to ride, easier to handle, easier to climb on, and the ground is so much closer when, and I do mean when not if, you eventually hit it.
Now, I have to comment on this. The only part of this you are right about is that they are sometimes
easier to get on and it usually
isn't so far to fall (unless they launch you, which they sometimes do
). In my experience, the short horses with short backs are the hardest
horses to ride because they are just so darn quick. A big horse, you can feel when they are about to spook/misbehave and have an opportunity to stick with them, even on a big spook where they spin and/or bolt. Those little short athletic boogers...not so much. They can be spooked, spun around, and bolted 30 feet the other way before you even realize they saw the boogeyman. It's all very cartoonish...where they leave you hanging in the air and it takes a couple of seconds before you realize that they're just...gone.
As for smaller horses being easier to ride and handle...that all depends on their training. I've known lots of 18hh draft horses that were infinitely easier to handle than most standard sized stock horses or even ponies.
That is exactly why it's so important to look for training and temperament instead of just size or breed.