There's 3 ways to buy a horse for racing in the U.S.
1. private sale - if you identify a horse you'd like to buy, and you can contact the owner, you just call and make an offer. If the offer is accepted you pay by bank wire transfer or cash to the seller, and you get a notarized bill of sale signed by buyer and seller, and you receive the original registration papers for the horse from the seller.
2. claiming from a race - the U.S. has claiming races at most tracks with a set price for each horse in the race. If you identify a horse to "claim", you must have deposited money on account at the track where the race will be held, and then you must submit to the racing office a "claim slip" properly filled out and signed by the buyer, no later than 15 minutes to post time for the race. After the gate has opened for the race, and if no other party also submitted a claim for the same horse, you then own that horse and must pick it up in the paddock after the race.
3. auction - many auctions are held in the U.S. and in fact all over the world for race horses of all ages. Each auction company has rules for how to go about bidding on a horse. Keeneland and Fasig-Tipton sales companies are 2 of the biggest auction houses for thoroughbreds in the U.S. See their web sites for details.