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Why was this downvoted? The swing analogy is horrible. There’s nothing physical that’s going to change when u play the game the exact same way as 04 05 06. The difference is u know what’s going on as opposed to not knowing where anything is. A good analogy would be playing Pokémon red again after beating the elite 4. The game itself won’t be any worse, it just will be easier and maybe less enjoyable if all you enjoyed is the aspect of not knowing where to go. But this is why pvp servers are magical, because each time you level a character, your leveling experience is different based on the encounters that happen while questing. In 05 U may have breezed through stv, but in classic u may spend an hour on a single quest solely because of the competition between factions on that particular quest.
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).
Each battle that you win will yield experience for each pet that participated. However, deceased companions will not receive any experience (so make an effort to keep them alive by swaping in a different pet). Experience gains are based on your pet's level compared to the wild pet. Defeating a higher level opponent will grant more experience than a much lower level one, but be ready for a tough fight!
Most people come here not to read walls of text, they look for the better-rated strategies that match their pet collections, copy&paste stuff they need to import into their addons and only come back if something goes horribly wrong and then mostly to find help in other peoples' comments who also struggled or to find a different team/strategy: I don't think many people wanna read about stuff that already happened to them, they mostly want to see solutions and it's the best way to have them in a well-optimized strategy&team already.