I've helped a lot of kids buy horses but the same rules apply to adults out shopping.
First, if you find yourself feeling uncomfortable or scared around the horse, walk away and keep shopping. Every person who has ignored that rule...including myself, I'm embarassed to admit...has been sorry. There are thousands of horses out there. There's no reason to start out being scared of one. ;)
Generally you'll find one or two breeds predominant in whatever area you live. Due to sheer numbers, those are the breeds you'll end up looking at. So my advice is to forget breed and look for a suitable horse.
Decide exactly what you will be doing with the horse, then look for one whose temperment, soundness, size and looks most closely suit him or her to your needs.
For example, if you want to ride over a lot of jumps, the horse must be 100% sound while if you just want to take some easy-going trail rides a few times a week, some minor physical problems are acceptable.
Finally, never, ever, no matter what buy a horse without getting it vet checked. Yes, it'll cost you some money but there are conditions a vet can discover that could end up costing you a whole lot more than the vet check did. Just keep in mind that there are also some physical problems that a vet check cannot discover but getting that check done is still a really good idea.
Oh, yeah...after paying the vet, listen to what he or she tells you. Don't worry, if the vet had bad news you ignore it and buy the horse anyway, you'll be paying that vet plenty more cash down the road. ;)