When you really like a horse, go ride him/her a few different times. To make sure this horse really fits your personality. Also take notice of his/her previous owners, are the stables clean, are they interested in the potential new home for their horse?
Make sure you find a horse that fits your training and experience level. Once you have found horses that do, try many of them out - not the first few that you find. Dont bw afraid if you dobt find one right away! Also make sure to ask lots of questions and bring your trainer or someone with a lot of horse experience along with you to help! Good Luck!!
Don't buy the first horse you try. Try out a bunch of different ones! And look at it realistically (like is a good match to your riding level, does he do what you want to do or will he need training, etc)