Throughout the game you will unlock different animals outside of your usual sheep and chicken, and gain the ability to build a friendship with them.
Unlocks fairly early in the game ( around 17th of spring ) when exploring Mount Whistle. You will get a cut scene with the dog, but to unlock it you need Super Gloves. The Super Gloves are among the gadgets possessed by Ryam, and to get them from him you need to grow and harvest three types of crops on your farm.
The cat is possibly unlocked by raising Doraemon’s friendship to three hearts. You will get a cutscene with Doraemon talking with the cat, and afterwards you get a mission to befriend the cat for him. The cat usually appears on your farm in the mornings, but sometimes wanders to the village.
The horse is unlocked by building the Horse Stable from the Lumberjack Shop.
The monkey is unlocked by raising your friendship with Sue to three hearts. Afterwards she will ask you to a picnic with her and you will come across some hungry monkeys in the forest.
The bear is unlocked by raising your friendship with Sneech to three hearts. You will then have to save him from a ‘bear attack’ and drive the bear away, but after the cutscene you can then find the bear from the nature around Plash River and befriend them.
The squirrel is possibly unlocked by raising your friendship with Big G to three hearts. You will unlock a cutscene in West Natura of the boys chasing the squirrel and you have a chance to save the critter from them. Afterwards you can meet the squirrel in the nature around the forest.
The rabbit you unlock by raising your friendship to three hearts with Vera. When you go visit her then at the Tree Roots you will find out that she has found a small rabbit that needs taking care of, and you can volunteer to help taking care of them as well.