The first step towards getting Epona is firstly obtaining the ability to play Epona’s song. As young Link, go to the Lon Lon Ranch and talk to Malon. Keep talking to her until she mentions something about a song. Afterwards, show her your Ocarina and she will teach you Epona’s Song.
Now comes the main part. As Adult Link, head back to Lon Lon Ranch, then to to the fenced area where the horses are kept. Ingo will be standing outside the gate.
Your first challenge is to learn to ride a horse. Ingo will offer you this service, at the cost of 10 rupees. Accept, and as soon as the mini game begins, you’ll be thrust inside the horse pen. However, do not mount any of the horses. Instead, play Epona’s song and Epona will come to you.
Once you are riding Epona, use her to jump over the fences in the horse pen – there is one small jump, and a higher jump. Press A to jump as you approach a fence. Once you’ve cleared both of the fences, go talk to Ingo. He will offer to race you in a horse race, wagering 50 rupees. Accept this challenge. Note that you may need to repeat the fence jumping challenge a couple more times before Ingo finally offers to race, I’m not sure whether it was intended or if it’s down to the game not registering successful jumps properly.
There are a total of two races to complete. The first race is fairly straightforward. Just use the directional buttons to steer your horse, and press the button to use up a ‘carrot’ to gain a short speed boost. Ingo will cheat and start slightly early, but this is no problem. Ingo will steer clear of the fence throughout the race, thus you can easily overtake him by using a speed boost carrot when a gap emerges. Try not to use
all your carrots at once. Once you defeat Ingo the first time, he’ll hastily offer to race you again. This time if you win, you get to Keep Epona! In this race, not only will Ingo get a cheap head start, but he will try and push you away. Use your carrots conservatively and when a suitable gap between him and the fence emerges, go for it. Once you win the race, Ingo will get angry and refuse you exit from the ranch!
Of course, now is the time to escape with Epona. Ride Eponda and build up some speed, then jump over a Fench around the ranch. You’ll land safely in Hyrule field with your new steed, Epona!