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).
That's not to say that everyone has an unrealistic vision of what vanilla WoW was like. There were already plenty of people on the WoW Classic forums pointing out that perceived bugs are just recreations of the original game. And as noted before, a flaw to one player is a key component of the original positive experience to another. WoW Classic will surely please plenty of purists in spades. But this goes to show that it won't be for everyone who has fond memories of the game circa 2006.
The vanilla WoW Horde leveling guide you see on this site was originally made back in 2006 by Joana (AKA Mancow, or FuriousPaul). The guides have been tweaked many times over the years to make things "faster". The leveling guides were made because of all the messages I got from people asking me how I was able to level up so quickly on new realms. I was first to level 60 on 7 different realms (3 with Mancow and 4 with Joana), even winning Blizzard's "First to Level 50" contest they held back in 2006.