The key to winning races is to be prepared for it beforehand. Before running/signing a race, you can view how long is it as well as the course route. It allows you to estimate how much stamina the horse will need to finish the race, and what stats are more important in order to win. However, the training of your horses is a the key aspect for winning races.
Breeding is possibly the most important aspect of the game. It allow you obtain better horses by combining different pedigrees. You unlock breeding by wining the 16 mile race, or by having a GIII horse. In the beginning of the game you have only two different pedigrees, which is not enough to win every race. By using breeding, you can combine two pedigrees to obtain a better horse. When you breed a horse, he will inherit the traits of his parents. A horse can have a maximum of five traits, although only four of the five can be upgraded. The traits in the colt (male young horse) are combined adding the traits of the mare (female horse) to the end of the traits of the stallion (male horse). All the horses that you can have are stallions (buying or breeding), so you must bring your horse to another ranch to breed.