A new quest will pop up once your first pet reaches level 3. It will send you to either Audrey Burnhep in Stormwind or Varzok in Orgrimmar. These trainers will offer a quest chain to battle specific Pet Tamer NPCs on each continent. Defeating each of these NPCs will reward a nice chunk of experience for your pets, as well a Sack of Pet Supplies (which has the chance to drop the Porcupette companion, among the always-useful pet bandages).
The simple truth behind all of this bickering and squabbling is that there really isn't a bad way to enjoy a game. You like the feeling of Classic taking you back to your young(er) days and that rush of falling in love with WoW for the first time? You like the slower pace of both combat and leveling? You like that the social aspects of the game take a front seat? Good, enjoy it! You prefer modern WoW and BfA with it's tons of improvements, smoother design and more player-friendly features? Great, enjoy it! It doesn't matter whether you prefer modern WoW or Classic or WotLK or whichever your personal high-point was (it was Burning Crusade and Legion for me), you're not wrong. You literally can't be. No matter what anyone says in all these heated discussions, they care about WoW, in whatever form they prefer it. So why can't this be enough of a commonality for us to have a civil discourse?