You can buy animals from the game shop. To do that, tap on the construction menu icon (the helmet icon located in the left corner of the screen):
In the menu that opens, check whether there are any animals available for purchase. If there is one, it will be highlighted with bright color, and the price will be indicated near it:
Tap on the animal to buy it:
After that, place the animal on any available spot on your farm and tap the green button:
In case you have reconsidered buying the animal, press the red button. The animal will then disappear and no coins will be withdrawn.
IMPORTANT: Having purchased an animal, you cannot cancel the purchase and get a refund.