You’ve had great advice here already, from experienced horsemen. Good luck, and happy shopping.
Though you can’t ride a horse that is for sale at the track, you can often watch them be ridden by one of the exercise riders on the track. Offer to pay for this, will cost ya $10 to $20. This is possible if the horse does not currently have a soundness issue that precludes him from being ridden at that moment. You can ask them to trot the horse for a while, and canter, to get some idea of how they move, and how tough they may be (or not be) to ride. You must have enough experience to be able to recognize what you are looking for when you see it. I like to see some natural carriage that I like, a nice disposition, a soft stretch with the front leg, and an active hind leg.
And yes, disposition is important, talk to grooms who work with the horse daily, information can be gleaned if you ask. That being said, I did buy a horse that I had watched and liked for several months, who had a disposition like an alligator at the track. Came at people with his jaws wide open over the stall door. But the groom told me that he was “just miserable here” (at the track), and he’s not really like that, doesn’t want to be here (at the track). So I bought him. He was a great horse, very talented jumper, but sensitive for sure. He bit people if he felt they needed it for an education, and he was usually not wrong in this judgement. But was very sweet if you had manners and were well behaved and fair with horses. Won a derby at Spruce Meadows (beat all the international horses who flew in from Europe) with my friend who bought him from me, made my heart swell with joy. I just rode him in the 4’6" ami divisions. “Sand and Surf”, and he never had his name changed.
If you want to be a horseman, eventually you have to make the decisions about what you like, and what you want to buy and attempt to reschool. The money you put where your mouth is, is the tuition you pay at the school of horsemanship. You learn as you go. You can take advice, you can ask for opinions, get coaching etc. But it’s your decision what horse you choose, and how you go about with the reschooling.
Step one, rebreak the horse… run him through basic training all over again, look for “holes” in his training that may have been missed or skipped. Start with ground work, handling, lunging, maybe even long lining. Fill any holes you find. Make a connection with him. Show him what you want him to do for you, it will be different from what has been asked of him previously. TBs are plenty smart, but if they have had race training, they need to learn a different carriage, different cues, and not to balance on your hands. They have to learn to carry themselves, like we expect riding horses to do. People who make mistakes about this, and skip this basic step are still riding racehorses and trying to get them to jump, and it never works out well. Make sure that the horse understands the difference that is necessary, to make the change from racehorse to riding horse. No leaning on your hand, and recognizing leg cues. Other than basic breaking, exercise riders fairly soon start shortening stirrups and don’t use a lot of leg cues when galloping. Some riders don’t know much about using leg cues or basic dressage (some do), and if the rider doesn’t know, the horses they ride never learn.
Good luck, and enjoy the process. People used to do this all the time, before the WB invasion. And many people had good success with it.