If someone sends you a friend request, a notification will alert you. Just click "accept" to add them to your Petpet Pals list. You can add friends yourself while roaming the park by clicking on the Petpet of the friend you want. This will bring up a menu that allows you to add the person to your Petpet Pals list. If you wish to manually add a friend, scroll down to the bottom of your Pals list and type their username into the input box there.