Easiest way is usually the simplest:
Have 2 Level 25 pets, grab a third you want to level. If said pet is Level 1, you're going to need to get it a few levels first. Easiest thing to do, is use the 3 dailies that give XP (Elwynn/Westfall/Redridge or Durotar/Barrens/Ashenvale).
Put your Level 1 wimp first in the lineup.
In the battle, make the level 1 go one turn then swap your Level 25 in and finish them off.
By the time you finish those 3 dailies, your pet should be ~Lv10 IIRC without the Safari Hat.
Now they're strong enough to actually level. Get a Lv25 Flying Pet with a Heal Spell (the Pterrodax Hatchling is easy to get and works awesome for this) and go to Jade Forest.
Many pets in Jade Forest are Aquatic (and thus weak against your main two abilities). Set your Pterrodax (if you're using that) up with Flyby, Blessing, Lift Off.
Go to the river where you see all of those turtles. These are all aquatic and that hatchling will absolutely destroy them fast. The hatchling also has a powerful heal spell, and an ability that allows you to dodge for one round. Using Flyby and then Lift Off pretty much guarantees a quick kill on anything aquatic as long as it isn't a blue. As far as I know, there's nothing he is weak against in Jade Forest (except the serpent hatchlings), so it should be very easy.
Anyways, the method is the same: Let the wimp take 1 turn, then swap him for the hatchling. Destroy the enemies. Watch him gain 900 XP per battle which is a guaranteed levelup until at least Level 16-ish, then every other battle gives a level until at least 21 or so when it might start to take a battle or two.
Edit: Do not try this in Krasarang Wilds -- there are millions of Jungle Grubs down there and they almost always use Sticky Goo their first round, preventing you from swapping your wimpy pet out.
Edited, Jan 10th 2013 7:35am by Lyrailis