While the box claims there are tons of animals including a cockatoo, the inside manual correctly states that there are SIX animals in this game. You start with three:

Guinea Pig
Most guinea pigs seem to be named Ewok!! I got four Ewoks in a single day once. These are easy to care for and easy to fix up.

Rabbit
Another starter pet that is fun and cute. The rabbits seem to get things stuck in their eyes a lot.

Parakeet
These budgies are NOT cockatoos! Sheesh :) They also look like morning doves when they sit on the operating table for some reason. Also, when they're in their cage, they NEVER perch. They just fly constantly. Hmmmm. And they're the only pet you can't pet!

Cat
Once you take some training, you can learn to care for cats. One is named Shiro, like the cat in the book! I thought that was great.

Dog
Dogs are rough to care for because they want to go for walks - and it takes a LONG time to walk the dog. You can only walk one at a time! Why not have a dog run where you can let them out to romp around?

Horse
One of the "features" of this game is that you can ride the horses. It is sort of fun, but I'd much rather ride the Zelda horses around vs dealing with this right-angle problem.

Pet Vet Walkthrough

Paws and Claws - Pet Vet Review