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You have to raise a pet to level 10. Capturing a pet 10 or higher does not count.
Capturing a level 9 (or lower) pet, and raising it to level 10 does work though.

If you already have one or two level 25 pets, then levelling more of the families you're missing can be easy by fighting high level opponents. You only need for your low level to survive one round, then put it away and kill with a high level. If you use both high levels, then deliberately kill one off so it doesn't subtract from the final exp. With this method you can usually get over three levels for your pet in the first battle (level 1-4), then two levels for the next couple of battles. By then you're almost there. It's over so quickly it doesn't matter too much what level the pet started as, as long as it's under level 9.

But if you don't have high levels and you're after the fastest possible solution, then you probably want to start with pets as close to level 9 as possible. Level 9 pets are found mostly in the Southern Barrens and the Cape of Stranglethorn, but not all families are represented there. In fact not all families have capturable battle pets under level 10. And, of course, if you want to level rares, then it may be faster to level lower level ones that you already have than finding new ones already at level 9.

Anyway, as far as I can tell, the ones that have capturable members at level 9 are:

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• Aquatic: [npc=61312] in CoS; [npc=61071] in SB
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• Beast: [npc=61318], [npc=61321], [npc=61320] and [npc=61324] (tricky to find) in CoS; [npc=62127], [npc=61325] and [npc=62130] (hard to get) in SB
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• Critter: [npc=61319], [npc=61317], [npc=61169] and [npc=61323] in CoS; [npc=61141] in SB
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• Flying: [npc=61314]s only show up as secondaries in pet battles in CoS (also level 7-8 in Northern ST). There are level 7-8 [npc=61322]s in Northern Stranglethorn and level 7-8 [npc=62177]s in Desolace
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The highest level members of other families (level 8 or lower) are:

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• Elemental: level 7-8 [npc=62182]s and [npc=62181]s in Desolace, then level 7 [npc=61703]s (hard to come by) in Arathi Highlands
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• Mechanical: level 5-6 [npc=62120]s in Stonetalon and then level 3-5 [npc=62119]s in Azshara
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• Undead: level 6 [npc=61758]s in Hillsbrad. Then 3-5s: [npc=62034]s in Ghostlands, [npc=61827]s in Bloodmyst Isle and Silverpine Forest, and [npc=61828]s in Bloodmyst Isle and Silverpine Forest
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As far as I can tell, there are no capturable battle pets in the level 2-9 range for the Dragonkin, Humanoids or Magic families. You'll have to level them from 1. I'm very happy to be corrected if I'm wrong though :)

There is a total of 10 pet families (Aquatic, Beast, Critter, Dragonkin, Elemental, Flying, Humanoid, Magical, Mechanical and Undead), so this achievement requires you to raise 10 pets, 1 for each family, to level 10 through pet battles.

Credit is given for each family of which you raise a pet from level 9 to level 10. This means first buying a caged level 9 pet, or capturing a level 9 wild pet (which will retain its level after capture) is the most efficient method for getting this achievement.

Before this first tier achievement is completed, the second tier achievement All Pets Allowed already starts tracking progress, so no credit towards that achievement is missed. This means it most effective to first level a team to 25, and then powerlevel the pets still required for this achievement to 10.