The Royal White Stallion
If you have seen any cutscenes so far in the game involving Zelda and Link, then you will know that Zelda rode on a beautiful, white horse. Talk to Toffa at the Outskirt Stable and he will tell you that a descendant of this royal stallion has recently been spotted at Safula Hill.
Once you reach Safula Hill, the white horse is easy to spot. However, this horse is incredibly skittish and once you mount it, is also difficult to tame. You will definitely need more than one wheel of stamina; ideally you have two full wheels of stamina. If you do not have that, you can eat food while taming the horse, just don't get carried away and forget to press the [+] button to access your inventory. You can also eat food that gives you extra stamina wheels.
To complete this sidequest, ride the horse safely back to the stable and register it. You can name it whatever you want. After that, talk to Toffa again while still riding it and he will give you the Royal Saddle and the Royal Bridle to dress up your horse with.
The white horse has its own unique Hyrule Compedium entry, so be sure to take a picture of it. Additionally, it has an extremely wild temperament so expect it to disobey you a lot on the way back to the stable. Remember to press [L] to soothe your horse and increase your bond. The horse has very well-rounded stats and will overall be a great addition to your registry if you love using horses. Just bear in mind that this horse cannot be replaced if it dies unless you go to the Horse God's fountain.