When you first get to Hyrule Field, head for the middle to Lon Lon Ranch. The only things you need right now are Epona's Song and at least 70 Rupees, both of which you have should. If not, travel back in time and go get it. Anyways, at Lon Lon Ranch, you see that it has a new owner, Ingo. Go and talk to him, then pay him 10 Rupees to ride a horse.
As soon as you appear inside the pen, play Epona's Song to get Epona to come to you, then ride her around a bit until time expires (you can talk to Ingo to quit riding if you want). Do this again, but talk to Ingo and he'll offer to race you for a 50 Rupees wager.
There will be two races against him, with the second one being harder. You do not want to expend all your carrots right away, or at all, since this will leave you without a boost for a few seconds. You want to try and pass Ingo around one of the corners and use a few carrots to get the lead on him, then using a carrot or two to keep ahead.
After winning both races, Ingo says he will give you the horse, but you won't be allowed to leave. He shuts the front gates to the ranch, leaving you stranded inside. What you want to do is get a running start and head towards one of the lower walls in the back section of the ranch. With enough speed, Epona will jump over it and you'll land in Hyrule Field. You now have Epona for the rest of the game.
OPTIONAL RACE
There's one optional race you can do now, but it doesn't get you anything worthwhile. Head back into Lon Lon Ranch after getting Epona and you'll see that Malon is now back in control. If you talk to her twice, she'll let you do a race, where you need to do two laps around the pen with fences in the way; this has to be done in under 50 seconds to pass. If you finish this, she'll deliver a cow to your house in Kokiri Forest. Remember that you can play Epona's Song in front of any cow to get free milk, which refills 5 hearts of health, and you get two uses per bottle.
