This guide will show you how to get Epona (your own horse) in Zelda Ocarina of Time Remake.
If you have played the original, it's pretty much exactly the same.
As a child, you learn Epona's Song from Malon, and then as an adult you use the song to call the grown horse, win a few races and she is yours.
Do not skip the child visit. If you reach adulthood without the song, you cannot get Epona until the Forest Temple is finished and you can travel back in time.
Here is the step-by-step guide...
Step 1: Hyrule Castle
Child era.
Advance through the main story until you wake Talon at the castle so the ranch opens.
Step 2: Lon Lon Ranch
Child era.
Talk to Malon in the corral and take out the Ocarina when she asks, to learn Epona's Song.
Step 3: Lon Lon Ranch
Adult era.
Once you are an adult, return to Lon Lon Ranch and pay Ingo 10 Rupees for a riding session. Play Epona's Song inside the fence and mount her when she comes.
Step 4: Lon Lon Ranch
Adult era.
Ride up to Ingo and talk to him from the saddle to race him. Win the race and then the rematch, then jump the closed gate to escape with Epona.

