Yay for you & what an exciting time! I got my first horse around 13 too.
If I could say one thing, it would be don't get a pony. You are quite young still and have a lot of growing to do and you'll soon be too heavy for a pony (and you are already quite tall.) I'm 175cm and my first horse was ~15.2hh, my friend is your height and her horse is 15hh.
I would be looking at something between 14.3 & 16hh because your first horse is a learning experience no matter how good you are at riding and it might be best not to give yourself too high a distance to fall! And personality/suitability ALWAYS should come before looks & height etc.
I made my 16hh horse look small & only now am looking at 16.2hh+ horses. Good luck