The Horse and Dog
The two pets you can receive in the game are a horse and a dog. They will both live inside the Pet Barn, which is just west of your house. Inside the Pet Barn will be 2 food troughs. You will have to put Pet Food in the food bins every day, even if you leave the horse and dog outside. The Pet Food can be purchased from Mirabelle's shop for 10 G each and there is no way to manufacture it on your farm.
When you first start out you won't have either pet. You will need to increase your friendships with other villagers before you will be rewarded with animal companions.
To earn the brown-colored horse, work on raising Chen's friendship level to almost 2 hearts (5000 FP). If the weather is clear, exit your house and Chen will greet you with a new horse. After you name the horse, Chen will place it inside your Pet Barn.
You can ride your horse around the island and outside of your farm. To mount your pony, stand next to it and tap on its back. To dismount you just have to tap on your charcter. You can ride the horse in and out of the Pet Barn.
If you leave your horse in an out-of-farm area overnight, it will return to your farm the next morning. The horse will be running around your farm land and you'll have to catch it if you want to ride on it again. The horse likes to run and doesn't stay in any particular spot for very long.
The horse festival is on Summer 23. The more hearts your horse has, the better it will fair in the judging. You will win if you have at least 7 hearts.
The dog comes from befriending Mirabelle up to almost 2 hearts (5000 FP) and then walking into the Forest area from W.Town when it is Rainy, Drizzly, or Snowy weather. You also need to have at least 1 poultry or livestock animal on your farm. You'll trigger an event where the animal shop owner is hunting for a dog wearing a scarf. When the two of you locate the puppy, Mirabelle asks that you take care of it on your farm.
If you leave your dog outside, and it has a lot of hearts, it will scare away the wild dogs that randomly appear on your farm. The wild dog's goal is to scare your livestock and poultry by startling them with its fierce bark, so having a dog of your own will even out the battlefield.
Protect your animals!
If the wild dog scares your animal friends, they will be grouchy the next day. Grouchyness can lead to sickness so protect your animals by building a fence around them in your crop field if you plan to leave them outside overnight. Use stones, sticks, or lumber to prevent the wild dog from accessing your creatures.
The dog can also participate in the Dog Festival on Winter 10. Just like the Horse Festival, the more hearts you have with your dog the better your chances of winning.