Some ideas:
The pets are sold at a pet store, and you can buy them.
You also have to buy a cage for each pet.
If you sell dogs or cats, graalians can also buy a leash to walk their pet.
In the pet store is food for each pet.
The pets have levels, the highest level is 10.
Every level, you have to feed it a certain amount before the next level.
Besides feeding, taking out the animal for walks or teaching it new tricks helps it gain XP to become a higher level.
The higher level it is, the longer you can take it out for a walk, and the less you have to feed it.
You can let your pet interact with other graalians' pets.
If its an animal such as a horse, pig, ape, or llama, the pet will take you anywhere. You open the map, and click to a certain spot. The animal will be as fast as a skateboard, and will take you to that certain area.
You can have only a certain amount of each pet.