I have bought horses using the internet, and by seeing a horse I especially like and asking the owner if it was for sale. To search on the internet, do a Google or Yahoo search for "horse for sale in _______(wherever you live)" and see what sites come up. Also, you can go directly to the horse site and do an advanced search for a certain price range, state, breed, color, age, height, pedigree....almost anything.
You can go to auctions to look for horses for sale, but your choices are extremely limited and you may get something that you hadn't bargained for. You're better off buying a horse you can "test-ride". You always want to ask the owner to ride it first, as you observe. Then you want to ride it. You also need a vet to do a prepurchase exam, or have a knowledgeable horse person with you to examine the horse and watch its movement at the very least.
Believe it or not, I have bought two horses by getting in my car, driving around, stopping at horse places, and asking "Do you have any horses to sell? I'm looking for a 2 - 3 year old gelding to train." Most horse people are more than happy to show you ALL of the horses they have, and they usually have one or two they'd part with. Both times I bought horses this way, they turned out really well. The owners weren't trying hard to sell them, and I didn't have to wonder "Hmmm...why are they selling this horse? What's wrong with it?"
Good luck horse shopping! Two good sites I like are www.liverystable.net and www.equine.com. Have fun!