Head to any tavern and ask the suspicious man in the back if there are any illegal horse races happening anytime soon. Head for the location given by him and ask the suspicious man in the new location the same thing. This will lead you into the racetrack.
Keep in mind to have a horse with a speed of at least 48+, since there are a total of 3 races, each with an increased difficulty in speed.
Do not pass the line before the count down at the start is over or your character will be DQ'ed from the race.