I bet the bounce rate on this page is going to be ginormous since being able to level a pet from 1 to 25 in four fights has so many prerequisites that its usefulness is limited to veteran pet-battlers. Literally: in order to do this right, players must have access to the Beasts of Fable quests at minimum, and even then players aren't guaranteed to have two of the items that make this possible: Pet Treat and Lesser Pet Treat. Both of these items are a must-have; they have chance to drop from reward bags players acquire from the Beasts of Fable quests and the weekly PvP pet battle quest.
Yeah, so it's kind of cheating---the treats will stack with each other, meaning you can have one of each active at any time to provide +75% XP gain. Throw on a Safari Hat for good measure and you're four fights away from raising that noob pet to max level (note: you may be able to do this without the Safari Hat, but why would you?). The order of these fights doesn't matter---in fact, any four pandaren master tamers will do the trick.
Here's what it looks like:
Fight 1 - Aki in the Vale of Eternal Blossoms - Level 1 to 14 (+6706XP)
For easy-mode, use: Anubisath Idol
While it is entirely possible for your level 1 pet to survive Aki's grasshopper's first attack, it's not guaranteed. The Anubisath Idol can nearly solo the three on its own thanks to Sandstorm and Deflection. To be safe, start with the Idol, cast Sandstorm and swap your leveler in for a round---they won't take any damage. Swap the Idol in and kill the grasshopper. Use Deflection on Aki's next pet's first move, then use Sandstorm again to remove its weather effect. Finish off the second pet. The Idol may die facing the third pet---En-Bee-Dee---just finishing him off with your third pet. A flyer will work best for this.
If you do not have an Anubisath Idol: Get one. Otherwise, any pet with Sandstorm will make this fight considerably easier---just use it like you would were you using an Idol. If you don't have a pet with Sandstorm, your level 1 pet can often survive one round from the grasshopper, provided there are no crits. The key then is to start with your leveler, swap it out and counter Aki's three pets with two from your arsenal. At the end of the day, the Idol makes this fight downright trivial.
Fight 2 - Nishi in Valley of Four Winds - Level 14 to 20 (+8510XP)Use a strong aquatic, as you're facing two elementals and a beast.
Nishi's no longer a pushover thanks to a change in patch 5.4. Siren, the pet she starts with, has been buffed. Solar Beam hurts, especially when you're also dealing with the pet's damage-increasing weather-effect. Avoiding Solar Beam will make this fight considerably easier. It should be the third ability Siren uses; I usually eat the Solar Beam, but reduce the damage it does by changing the weather before Siren casts it. Do what works for you.
I normally wait until Nishi's second pet dies to swap in the leveler, but this isn't necessary. I recommend finishing the battle with a mechanical pet.
Fight 3 - Yon in Kun-Lai - Level 20 to 24 (+6706XP)
For easy-mode, use: Emerald Proto-Whelp
Apply the Proto-Whelp's damage-reduction ability and you'll burn through Yon's first two pets before you know it. Otherwise, know that Yon starts with a flyer whose main ability is a multi-strike attack---if you have any damage-reduction buffs, just like when using the Anubisath Idol with Aki's grasshopper, these won't damage you. If you aren't using a Proto-Whelp, your first pet usually dies partway through facing Yon's second pet---here I'll bring out the leveler to hopefully finish it off---remember, your leveler is 20 now and thus can do decent damage as well as take some hits---then swap in a mechanical to finish the battle.
Fight 4 - Your Choosing/Burning Pandaren Spirit
For easy-mode, use: Pandaren Monk and Anubisath Idol
I went to the Burning Pandaren Spirit just because I have decent pets to counter the burning spirit's and it was closest. Really---if there is another tamer in Pandaria you're comfortable with, go for it. Any win will net you the remaining experience you need to get to max level.
For this fight, start the monk and use Jab. Then use Focus Chi---your opponent will use Lift Off here. Finish the pet with Fury of 1,000 Fists. Missing this attack can complicate the rest of this fight. Assuming you've killed Burning's first pet, continue to damage the next pet until your monk dies. Bring in your idol and finish the second pet using Crush while applying Stoneskin and using Deflection to counter the elemental's one devastating attack---once you time this right you'll be using Deflection the round after it comes off cooldown. The third pet here is a flyer with a multi-strike attack which gets completely negated by your idol's Stoneskin. I usually drop the third pet low enough to where I can bring in my leveler for the last few hits of that fight.
Congrats! Now wasn't that easy?