This page explains how to get Epona in Ocarina of Time so Link can ride on horseback throughout Hyrule.
After you get the Master Sword in the Temple of Time, you can rescue Epona. But in order to get her, you will need at least 60 rupees, and you need to have learned Epona's Song from Malon in Lon Lon Ranch in the past.
Where to Summon Epona
Once you rescue Epona, you will be able to summon her anywhere in Hyrule Field, Lake Hylia, Gerudo Valley, and Gerudo Fortress, making travel around Hyrule much easier. You can also get Biggoron's Sword after you rescue Epona.
Step One: Practice with Epona
Go to Lon Lon Ranch as adult Link. Talk to Ingo by the entrance to the corral, and when he asks, choose to ride a horse. Once inside the corral, play Epona's Song, and Epona will come over to you. Climb into the saddle.
To ride, use the control stick to move. While moving forward, you can use a carrot to speed up. You only have six carrots. If you have some carrots left, you will gradually regain carrots until your meter is full again. But if you use up all of your carrots, there will be a delay, and then your meter will be refilled. If you want to go quickly, leave at least one carrot in your meter to avoid that delay.
Ride around the corral, and jump over the middle of the fences to get a blue rupee for each fence. You have to speed up to get over the taller fence. Ride around until Ingo stops you. Ask to ride again, call Epona, and climb up. This time, get close to Ingo while riding Epona, and talk to him. He'll wager 50 rupees that he can beat you in a race. Accept the bet.
Step Two: Race Ingo
Right at the start of the race, use some carrots to try to squeeze in front of Ingo, but don't use all of your carrots. Use carrots carefully to maintain your lead. If you're in the lead, don't use any carrots. If Ingo starts to come closer, use one or two. Make sure to have an almost full carrot meter as you get close to the finish line. If you lose, you will need to pay 50 rupees to race Ingo again.
After you beat Ingo in the first race, he will get upset and bet the horse on the race. This second race is harder. At the start of the race, you probably won't be able to squeeze in front of Ingo this time. Just use enough carrots to get right behind Ingo, and stay close to the inside track, but be careful not to touch the fence or it will slow you down. Try to get in front of Ingo without using all of your carrots. Ingo speeds up just before the finish line, so try to keep your carrot meter as full as possible before you get close to the finish line.
If you lose, just keep trying.
Step Three: Escape from the Ranch
Once you win both races, Ingo lets you keep the horse, but he traps you inside of the ranch. You can get out by having Epona jump over a wall. Just turn to face the nearest outer wall that has spikes on the top, and gallop toward the left side of it, using all your carrots. You'll jump over the wall and leave the ranch. Now you can call Epona from Hyrule Field, Lake Hylia, Gerudo Valley, and the Gerudo Fortress. Now that you have Epona, you can get Biggoron's Sword.