First quest is to go out and defeat a wild pet. Once you've done that, he gives you a quest to talk to Erma, Goldshire's Stable Master. Turns out that Stable Masters are now Pokemon Centers, and can heal all of your pets for free! After healing, you can hand the quest in. You then get Revive Pets as a reward!
So it looks like there will be three ways to heal your pets: Stable Masters, the Revive Pet spell and the Battle Pet Bandage, which I have no idea how to get!
