Children's Week (ID 172)

FYI: You must talk to the Matron first. She has an option on her gossip window, "Well what do you know, this is Children's Week! What can I do to help?" After using this option, the quest should be available.

I initially went to the Matron at level 8 and was unable to pick it up, went back at level 10 and I was able to accept the quest.

When the orphan is spawned, he gives you several quests; the final quest in the chain is
A Warden of the Horde
http://www.wowhead.com/?quest=5502
which has the rewards.

Use new vulpera or mag'har characters to complete the Orgrimmar quests in less than six minutes.

Vulpera/Mag'har for Pet Acquisition

1. 
   
2. Create new vulpera or mag'har character (not class trial), enter world.
3. 
   
4. Fly to orphanage, pick up Children's Week, summon orphan, get the three quests
5. 
   
6. Fly to Azshara rocketway, complete quest
7. 
   
8. Use [item=205458]
9. 
   
10. Fly to Red Rocks (if DMF is active, use [npc=55382] to teleport to Mulgore), complete quest
11. 
    
12. [item=6948]
13. 
    
14. Fly to Embassy, complete quest, pick up two new quests
15. 
    
16. Buy [item=69233] from [npc=52818] behind Embassy
17. 
    
18. Find [npc=52809] in the Drag, buy [item=69231]
19. 
    
20. Use Dragon Kite 2-Pack, dismiss orphan*
21. 
    
22. Re-summon orphan, use Cone of Cold
23. 
    
24. Turn in quests, get new quest, buy [item=69057] from Blax
25. 
    
26. Turn in quest, buy the pet you desire.
27. 
    
28. Repeat with new vulpera or mag'har.

I've completed this in 5:30. There's no point in using class trial mages for this - it's just as fast using vulpera/mag'har, and class trials can't use [item=241215] anyway.

You can shave off ~30 seconds if you completed Wailing Critters from the Wailing Caverns pet dungeon. It unlocks an account-wide teleport to WC. After Step 4 ([item=205458]):

• 
  
• Visit the Command Board in Valley of Honor, select Broken Shore/"Fight the Legion"
• 
  
• Talk to [npc=4311] outside Orgrimmar, take the skip
• 
  
• Walk into the Dalaran pet shop, ask [npc=121602] to teleport you to Wailing Caverns
• 
  
• Fly over the ridge to complete the Red Rocks quest, Hearthstone back to Orgrimmar and proceed with the rest of the steps
• 
  

[item=69648] is currently the only pet you can't get from this method; you need to complete the Shattrath Children's Week quests for it.

*Can't use Cone of Cold while Dragon Kite dialogue is happening. Dismissing your orphan stops the dialogue and saves you a few seconds. It also stops your orphan from running around with the kite.