Getting Epona is one of the first things you can do as Adult Link. I’d advise doing it either as soon as you become Adult Link, or after completing the Forest Temple (as then you’ll have the Bow, useful for horseback archery).
You first need a wallet full of rupees, and Epona’s Song (obtained by showing Malon your Ocarina after waking up Talon). During the day go to Lon Lon Ranch and ask Ingo if you can ride the horses. It’ll cost 10 Rupees.
Once you’re inside, use Epona’s Song to bring her over and then just jump on. Ride around the field and jump over fences, then talk to Ingo, who’ll challenge you to a race. If he doesn’t, try talking to him a few times. He ask you to wager 50 Rupees in the race. If he wins, you’ll lose the money, so you could actually need a lot of Rupees.
If you win, he’ll go crazy and challenge you to another race, promising you Epona if you win, but he’ll close the ranch if you do best him twice. Ride Epona to the fence on the Ranch-house side of the field, and speed up. With any luck, Epona will jump over it, and you will be in Hyrule Field.
Congratulations, you now have completed the Epona Quest! Ingo will give Malon the ranch back and she’ll fall in love with you and whatnot, but that doesn’t matter, it’s another story.
When riding Epona, six carrots will show at the bottom of the screen. Each of these represents a speed boost, which you can use by pressing A. You need a couple of carrots to jump the larger fences, but during races make sure not to eat the sixth carrot, because Epona will lose speed needed to jump even the smallest of fences. Carrots regenerate over time, so try and keep a happy medium, building up your stock when you’re ahead, and using them effectively when you’re behind.
Also, competing riders tend to stick to the inside track, so if you’re ahead you can block them from overtaking you by also using the inside.