Use Buy a Horse to browse available horses and filter the list to find what suits your stable.
You can also try to claim a racehorse from an eligible claiming race. The claim price matches the race level. You need an available Main Stall, and when several players claim the same horse the successful claim is chosen at random.
Auctions are another way to purchase a horse. Premium membership is required to buy or sell through auctions. Place a bid on an active listing and watch your notifications in case another player outbids you.
You may also create a horse with no ancestors in its pedigree by spending EQ Credits, or accept a private offer sent by another player.