Once you have narrowed down your search, ask the seller if it is acceptable to have a vet look at the horse. If they say NO, chances are you wont want it-there is probably something wrong with it Get an equine veterinarian to check the horse over. You can have different levels of prepurchase exams done by the veterinarian, at varying cost. Your purpose for buying and purchase price will determine the extent of the prepurchase exam. Your veterinarian will be able to help you decide how much work he should do in determining the physical suitability of the horse for your needs.
Many people price their horses allowing some leeway for negotiating.
Ask if the horse comes with any equipment. Most only come with a halter but some will come with all the equipment if the owner is no longer interested in horses. Ask if you can take the horse for a one week trial. Some people will allow you to take the horse for a trial period in order to see if the horse suits you and vice versa. Discuss liability during this trial with the owner and get any agreement in writing.