[FAQ] Pets
NEW! How do I get cats?
You need to build the Pet Store on the town map, and then tap on it to enter the Pet Store screen and select the cats tab in the bottom of the screen. There are 6 cat breeds to choose from.
Once you select a cat, you need to assign it to a Sim. It will then move into that Sim's house.
NEW! How do I purchase accessories for my cat?
Go to the Home Store and select the Pets tab.
If the item has a cat icon on the bottom right corner, then it is a cat accessory.
NEW! How do I get Fish Tanks?
Tap on the shopping cart icon to go to the Online Store.
Go to the Specials tab to purchase either the Tropical Fish Bundle or the Goldfish Aquarium.
NEW! I purchased Fish Tanks, where do I find them?
Go to The Home Store and select the Pets tab, you will find the Fish Tanks in there.
How do I buy a pet?
To buy a new pet, you must have a Pet Store built on the Town Map.
Select the Pet Store, choose a pet and select "Buy".
A scrolling window will appear with all of your current active Sims.
Choose which Sim you would like the pet to go home to, then confirm your purchase.
How do I change which house my pet belongs to?
Changing which family looks after a certain pet by selecting the pet, choosing "Relocate" and then picking which Sim you wish to look after the pet.
How do I get rid of my pet?
The "Put up for adoption" option is available in a pet's "Relocate" menu. Use this option if you no longer wish to delete the pet.
Source: https://help.ea.com/article/the-sims-freeplay-pets-faq