Pets were added to Hay Day in an update, but the game has a unique approach towards them - you have to save up 'Pet Vouchers' before you are able to get one. Pets include Cats and Dogs.
You can get pets by purchasing them. Dogs can be bought after Level 20, while Cats start at Level 21. You require Pet Vouchers as mentioned above, which are given out when you complete boat and special truck deliveries. You can read more on how to get Pet Vouchers, below.
Before you are able to get a dog or cat, you first have to purchase a Dog House, or a Cat House. Once you have them, they will walk freely around your farm. They do require some intervention, however, and you will need to be willing to feed them once they get hungry. To do this, you simply tap on the house that you built for them and fill the bowl either with bacon or milk, for the dog or the cat respectively. You can fill the bowls whenever you like - the pets will only eat once they are actually hungry.
Having eaten, your pets will want to sleep. To get your reward for feeding them, tap them to wake them up. If you have a lot of pets, use the whistle to wake them all up together and get the reward all at once.
You are limited to 2 Dog Houses and 2 Cat Houses on your farm. Each pet house can have a maximum of three pets. This means, if you so desired, you could support an impressive 12 pets (with 6 dogs and 6 cats) on your farm.
Use the pet whistle to control your pets. Blowing it will wake them up, allowing you to collect your reward, as well as to call them back to their houses if they've been wandering.