Instructions
Get your horse by visiting the Green Ranch as soon as you can. Leave your farm and head south (down) at the intersecting roads to reach the Green Ranch.
Get the animal brush from Rick's shop in town as soon as possible. The brush costs 500 gold, but it's worth having because it works on all animals and being brushed makes them happy.
Whistle for your horse every morning by pressing the Left-C button. It will run to you.
Equip the animal brush and use it on your horse. It helps to talk to the animal as well by pressing the action button.
Ride your horse when it is old enough to be ridden. You can tell it's old enough because a saddle appears on its back. Ride it by getting close and pressing the action button.
Repeat steps 3 through 5 every day (game time). If you work hard enough and pay attention to your horse each day, you can enter and win the horse race. It may take more than a year of game time before your horse is suited to win the race, but he can enter anyway so long as he's matured (has a saddle).