Aw, Isn't It Cute?

For the 2009 Children's Week event, I would suggest everyone not to pick the piglet pet as a reward due to the recent swine flu outbreak. If you must, wear a mask.

To obtain one of these pets, complete the following quests given by your orphan (this is the Azeroth version, thanks AnrDaemon for posting the Outlands version below):

1. 
   
2. Spooky Lighthouse
3. 
   
4. The Stonewrought Dam
5. 
   
6. The Bough of the Eternals
7. 
   
8. You Scream, I Scream...
9. 
   
10. Jaina's Autograph
11. 
    

The quest, A Warden of the Alliance becomes available after completing You Scream, I Scream... & Jaina's Autograph. You can choose one of the pets as the reward for this quest

Edits: adding links & improving look.

Horde version:
First quests available

• 
  
• Lordaeron Throne Room
• 
  
• Gateway to the Frontier
• 
  
• Down at the Docks
• 
  

After completing these, 2 new quests are available:

• 
  
• Cairne's Hoofprint
• 
  
• You Scream, I Scream...
• 
  

And finally

• 
  
• A Warden of the Horde
• 
  

If you have turned it both Shattrath and Org/SW orphans, but forgot to complete achievements, you can still ask the NPC to give you another Orphan Whistle