You won't be able to get Epona until Link is an adult, but there are a few things that must be done when Link is a child. After meeting with Princess Zelda for the first time at Hyrule Castle, go back to Hyrule Field and go to Lon Lon Ranch, which is in the middle of the area. Go to Malon, who is standing in the middle of the ranch and talk to her. Show her your Ocarina and she will teach you Epona's Song.
With that over, you need to wait until you become an adult. When you visit Lon Lon Ranch, you see that Ingo has taken over. Talk to him and he'll offer to let you ride a horse for 10 Rupees. Take him up on his offer and then play Epona's Song to get her to come to you. Ride around until the time limit expires, then do it again. This time, instead of waiting for the time to expire, talk to Ingo and he'll offer to race you for 50 Rupees.
The first race isn't too hard, but you have to make absolutely sure you don't use up all of your carrots. Use one at a time and wait for it to replenish before using it again. Try to go on the outside and then as you round a turn, go inside and pass Ingo up.
After the race is over, Ingo can't believe you won and wants to race you again, with the horse being your prize if you win again. This second race is harder, but if you follow the same tips as the last race, you shouldn't have too much trouble winning. One extra tip is to make sure you block Ingo's advances if he tries to pass you.
Ingo can't believe you won again and he isn't happy that you used Epona, as he was going to give it to Ganondorf as a present. He locks the gates and you can't get out under normal circumstances. What you need to do is ride towards one of the walls in the back of the ranch and if you have enough speed, Epona will jump right over it and you'll be in Hyrule Field again. You now have Epona to use at will for the rest of the game!
