Petting zoo is a good place for your Pets to hang out and get you some rewards! 
First of all, open Build menu and build Petting zoo.

Then open Petting zoo and pick the Pet. But note, that you can pick a Pet only when it is fed! Here you can see the resources you need to start the process and the souvenirs you can get.
   Then, after some time you will get a reward! 
   You can find them later in Build menu and open them.   It contains resources that are useful for building your island!