The very first pet you get in Story of Seasons: Friends of Mineral Town is a pony. After you look after it for about three full seasons, it will turn into a horse, and you can use it to get around town at a much quicker pace. If you want to get more pets, you will need to unlock Van, the Traveling Merchant.
You can only get more pets when any pet you already own has at least 8 hearts in their friendship meter. To check your pet’s friendship level, open the menu and go to the Relationships tab, then pick pets. Here, you will be able to see the individual friendship level of any pet you own.
To increase your pet’s friendship level, you should interact with them every day. Telling your pet you like them and giving them a brush down each morning is a great way to grow your pet’s friendship level. You can also buy a treat from Van for 1000 Gold each that will boost your friendship level; if you don’t want to wait to get a new pet, you can speed boost your pet’s friendship level this way.
Being better friends with your pets will also mean they are better at playing fetch with you. You can purchase a ball or a Frisbee from Huang, who lives in Zach’s house on the beach, and you can then play fetch with them when they are fully grown. You can buy the ball for a 100 Gold, and the Frisbee for 5000 Gold. You can use the ball to play with them when they are younger, and this will also help to increase your friendship level with them.