For the horse or pony thing, it depends on your personality! I learned how to ride on push button ponies when I was little, then started riding nasty ponies, then I switched over to horses, and now I'm looking for a pony again. Can you keep taking lessons at your barn before you decide to buy? That way when you go to look you'll be sure what you want height and discipline wise, and you won't be scrambling to buy or sell if you change your mind in the near future.
And quarter horses and ponies are known for their (usually) easy going temperaments, but it's not really fair to single out breeds because there are always exceptions. You just have to have a keen eye and always be on the lookout for a good match.