The following animals will become unlocked when you reach the indicated stage of the game.

Unlock Alpaca:
Unlocked in year 2 Fall.

Unlock Jersey Cow or Calf:
Unlocked in Year 2.

Unlock Owl:
Unlocked after the 1st tunnel upgrade.

Unlock Shetland Horse:
Unlocked after you complete the three tunnel upgrades.

Unlock Silkie Chicken or Chick:
Unlocked in year 2.

Unlock Suffolk Sheep or Lamb:
Unlocked in year 2.

Unlock Thoroughbred Horse:
Unlocked after the first tunnel upgrade.