First you'll need the right number of pet cards for the pet in order to add it.

You can get pet cards:
1) in packs from the pet store, which you can access through the Bank window or your Pet Room by tapping on Add;
2) from special offers.
Pets will be available after reaching level 15 at the Sakura restaurant.
How do I get into the Pet Room?
You can reach the Pet Room via your wardrobe. To do so:
1) Tap on your avatar.
2) In the lower wardrobe menu, find the paw-shaped icon. When you tap on this icon, you go to the Pet Room.

You can also access the Pet Room by tapping on the pet in your restaurant.