(If going to the PTR) Cage all your level 25 pets (uncheck "Can't Trade" to filter the rest out), all of your level 14-17 pets, and the level 1s from Step 1. Now transfer your character to the PTR. Remember that you will have 0 pets on the PTR and must learn Battle Pet Training and buy your first level 1 from the Tamer before you can learn anything. After you get your first 25 with this guide, you will suddenly have a massive army of 25s when you learn the ones you caged.
Step 1. Get any rare-quality level 1 Mechanical pet (you'll see why), preferably 2-3 of them. Examples: Clockwork Gnome, Darkmoon Tonk, and Darkmoon Zeppelin. If not going to the PTR, just create a level 1 Gnome on some Latin American/Brazilian realms and capture 3 different Fluxfire Felines (uncommon or higher). They work best, but you can use other stuff too.
Step 2. Level your level 1 Mechanicals to level 4 in your starting capital while doing your introduction Pet Battle quests (up until the one that makes you fight a tamer).
Step 3. Battle your first tamer. You should easily beat her with only using one pet, so do not switch to other pets for no reason. After the fight, your main pet should be level 5. Now fly to your second tamer and do the same. Your main pet should now be level 6 (though if you had to use 2 pets for that tamer and it isn't quite 6 yet, it's still fine).
Step 4. Go to Deadwind Pass and battle a level 17 Arcane Eye. Hopefully yours is Poor or Common quality, if not you may want to forfeit depending on your pets. With a Fluxfire Feline, you can take anything because it is just that freaking good (my level 6 Fluxfire Feline beat a level 18 uncommon Arcane Eye 1v1), but otherwise just be careful. If you fail, just use a bandage from one of your two pet reward sacks (smrt, huh). Anyway, capture the Arcane Eye in any way possible. Mechanicals are resistant to Magic damage and the Eyes tend to Eyeblast spam mostly (nothing you can do when they Gaze, just suffer the lost turn).
Step 5. Capture more Arcane Eyes. MOAR AND MOAR EYES! I would get 3 level 16-17 Uncommon or higher Arcane Eyes under my belt if I were you, but if you can't be picky (i.e. your CRZ is unusually low on them atm or something) it will still work. Your lineup should be straight Arcane Eyes and they should use the build Focused Beams, Drain Power, and Interrupting Gaze. Use Interrupting Gaze on CD and spam Focused Beams; use Drain Power if the enemy is stronger than you or has a good move coming up.
Step 6. Go to the Dragonblight and find one of the huge dragon skeletons out in the waste areas. There are at least 6-8 Dragonblight Condors at each skeleton on a super fast respawn. Engage the packs of level 22-23 Dragonblight Condors and back out immediately if you feel you can't take the adds. You want a group with 1-2 Poor or Common quality pets and preferably 2 Flying pets (your Arcane Eyes do bonus damage to Flying types). Capture at least 2 Uncommon or better Dragonblight Condors in this way before moving on to the next step (you can leave the third slot as an Arcane Eye to deal with Flying types if you wish). Remember you can use the nearby Stable Master at Wrymrest Temple for easy healing.
Step 7. Go to the Vale of Eternal Blossoms. Armed with your new army of Dragonblight Condors (moveset Slicing Winds, Adrenaline Rush, and Lift-Off), fight the roaming bands of Eternal Striders that border and sometimes cover the lakes and rivers in the zone. Five out of seven of the pets in the zone are Aquatic (weak to your Flying type Moths) and the other two types of pets in the zone are Flying (weak to your Magic type Arcane Eyes), so most of the time you will fight 3x Aquatic packs or 2x Aquatic 1x Flying packs. You can grind from level 23 to level 25 here in 10 minutes, easily. As always, use the stable master nearby at the capital if you need it.
You win! Now, hopefully more people will jump onboard the PTR bandwagon since they can have a lot of their favorite pets there now at level 25 in no time at all (Anubisath Idol, Magical Crawdad, Emerald Whelpling, etc).
TL;DR version: Level Mechanicals to level 5, use them to capture level 17 Arcane Eyes in Deadwind Pass, use them to capture level 23-25 Garden Moths in Jade Forest, grind with your Garden Moths against the Eternal Striders in Vale, win.