To get animals you have to build the Barn or Coop and reach Brownie Ranch. Barn- 16 Material Stone, 18 Lumber, $3000. Coop- 10 Material Stone, 10 Lumber, and I think $2400. Each take 2 to 3 days to build once Dale is commissioned. Brownie Ranch is unlocked in Fall or when you get to Level 3 with your Hammer, whichever comes first. If you get to level 3 Hammer: Go to the Rock blocking Brownie Ranch, Equip the Hammer, and push A or "Check".
Brownie Ranch Starts out with Chickens $2100, Calves $4500 (Cow), Lambs $4200 (Sheep), and Silkworms $2850. But they later get Goats $3600, Horses $6300, Ostriches $13500, and Ducks $2400.