Things you need before you can get Epona the horse: Epona's Song, 60+ Rupees (I recommend more rupees that 200).
First go to Lon Lon Ranch. Go talk to Ingo who should be standing by the big fence at the white line. Talk to him. Tell him you wanna ride. While riding, jump some fences and ride around. Do this until your time is almost up. Go talk to Ingo while your on the horse. He should be inside the pen now. He might ask you about racing him for 50 rupees, but if you talk to him and he ask you if you want to quit, keep riding until the time is up. If that happends, ride again and talk to him any time during your riding and he should say it. Anyway, if he asks you if you want to race him, accept it. If you beat him, he'll say he'll race you one more time and if you win, you can keep Epona. Race him again. This race is a little more harder than the first. When you beat him for the second time and he says you can keep the horse but he says he is never going to let you leave Lon Lon Ranch and closes the exit fence. Jump the fence and you have Epona the horse!
Tips on beating Ingo: Those carrots are for speed. 2:Don't use all your carrot things up. Wait until there is one or two carrots left and wait for it to re-charge. 3: Put your horse in front of Ingo's and he'll stay behind you.