Hmm. Fairly broad questions. I'm assuming you'll just be looking for a riding companion, then?
Get a nice QH, Morgan, or Paint.
Sorry, I don't agree with this statement in it's barest of forms. When you're searching for your first horse, breed shouldn't matter in the least, UNLESS you are looking specfically for a type of horse, then you search for all the qualities you'd need or want. The only time I think breed should ever matter is if you're breeding.
I've seen some Paints and Morgans that would make the craziest Arabian look sane, and some Arabians that make Paints or Quarter Horses look like psychotic skitzos. While some breeds tend to be hotter than others, it all comes down to the time and training put into that individual horse and by limiting yourself to those three breeds, you'll be missing out on some fantastic horses.
Appy's, although far from my favourite breed, tend to be very verstalile horses but the ones I've seen can be pretty hard headed, but a broke one I wouldn't imagine you having too much trouble with.
The only concern you'd have to keep in mind is eye problems and cancer due to genetics, though that's not overly common, though not overly rare.
If you're looking for a good riding horse, first things I'd start off with are ages and confirmations. If you're looking for something that's going to last you years to come, I'd probably start around the 6-8 age range because, generally speaking, they've got a solid foundation of training and are still young enough to go and go and go.
Legs are important. Nothing too straight in the shoulder or pasterns or hocks. Again, you're not looking for halter conformation you don't want anything too straight or crooked as it can limit their ability to perform and go.
Temperment and personaility matches are pretty much required. If you see a horse you think you like, go out and meet him. He could be gorgeous and sound great, but if you two don't get along you'll both be miserable. They all have such individual personailities we can get along with them like we do other people, hence why some people are drawn towards some breeds and not others. I think a few meetings with potential buys are very important.
Another thing I'd also reccomend it buying from someone reptuable. Don't go to a sale, or to someone backyard breeder who's got a great riding horse for $200 because odds are, the only thing that horse is good for is meat. Although the horse market is slow right now, good horses aren't really cheap and if you buy from someone who's honest and trust worthy, you'll more than likely find them to be more willing to help should any problems arise in the future with your new mount, but more likely than not, you'll probably NOT find any dishonest problems later.
It's a long tricky process but I hope you find your right girl or boy when they come along! :)