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).
When it comes to levelling the Priest Class from 1-60 fast, XP grinding locations is only half the battle. You will need to make sure your weapon does enough DPS so that you can clear these mobs fast enough, it should also be noted that monster should be around your level to make sure to gain the 100% XP Reward. The Level your character should be before you start that specific location will be indicated to left in a Blue Color for Alliance [Lvl 35-39] and the Orange Color for Horde [Lvl 35-39]. The Mobs you are looking for in that specific area names are Bolded [Mob Name].
Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.
Each pet has 3 abilities it can use during a battle, but a total of 6 to choose from. They will also have a total of 3 active spell slots that show which abilities can be used during a Pet Battle. The first spell slot will already be unlocked, but the second and third will only become available once the companion has reached a certain level. Other spells will also unlock after reaching higher levels.
Step-by-step class guides are on the way!  The 1-12 Mulgore Guide is the first guide to get all the individual class steps.  At the top of the guide, simply select which class you are playing and the page will dynamically update all the steps in the guide for that class (both text and images).  The rest of the guides will be slowly updated overtime to include all the class steps, but you can see how it's going to work with the 1-12 Mulgore.  I want to thank crazyK and his Placeholderguild for helping me out with the individual class content.  They are allowing me to get the 1-60 Alliance guide done while also getting all the class steps done at the same time, so everything can be ready for Classic launch (hopefully).

FYI: If you’re returning to WoW after an extended absence prior to patch 5.0.1, it’s possible that you’ll have more than the maximum pet limit and/or more than 3 of a kind. However, this will be the only time that you’re allowed to surpass these limits. You will have to pare down your collection, eliminating your duplicates, in order to add new pets.
With the Classic beta now out it seems every related article somehow manages to spark the eternal war of "Vanilla was the best WoW sucks now" and "lol nostalgia goggles, Vanilla sucked, enjoy your two weeks of Classic". I have to say, even though I understand the principles behind the battle and the reasons people behave and talk this way... I actually REALLY don't get it on a deeper level.