What to look for when buying a horse
1.) Temperment. Is horse generally calm, uptight, unable to perform a flat footed walk, bolting, etc? Horse should be well behaved on the ground (when you're walking them) AND when you're riding them.
2.) Training. Is horse trained/experienced in what you want to ride? Trail horse? Ask to ride him on a trail - if possible alone and with other horses. Jumping? Jump horse or have your trainer jump the horse. Etc... - in other words make certain your horse is trained in the discipline you want to ride.
3.) Health. YOUR vet, NOT sellers, should pull bllod work the day you go to buy (maker certain you ride that day) to ensure horse is not drugged up to be quiet nor drugged to hide lameness. Have your vet perform basic checks on: eyes, ears, legs, hooves, lungs.