1.) Make sure you have gotten Epona's Song From Malon as a child or you won't be able to do any of the following.
2.) Enter lon Lon Ranch and go talk to Ingo who is standing at the Corral Gates.
He'll offer to let you ride the horse for 10 Rupies. Agree. As soon as you get in the stables, play Epona's Song. She'll coming running to you. Get on her back and try to impress Ingo by jumping a few hurdles. After, you've jumpes one or two hurdles, go back to Ingo, while on the horse, Z target on him, and talk to him. If you succeeded, he will say that you are becoming a good rider and will challenge you to a race. Agree to the race.
3.) THE RACE!!!!
When you the race starts, start use up all your carrots except the last one right away. It's very important to get ahead of Ingo in the very begining of the race. Continue that pace and what ever you do, DON"T USE THE ALST CARROT!!!It will slow you down completely. He will be angry if you win the race and then will challenge you again. Use the same stratagy.
4.) Leaving Lon Lon Ranch with Epona.
If you win both races, he will give you the horse but says he'll never allow you to leace the ranch. Simplet jump over the gate that he locked, or head towards the back of the barn and jump the back wall. Epona is yours!!!! And trust me, you'll apreciate at how much she cuts back your time.