Please refer to the 2007 posts on alliance and horde for how to do the pre-60 quests: Turtle, Pig, Rat, or 5g.
Level 60 required.
Start in easternmost part of Lower City at the Orphan Matron, who will give you the orphan whistle.
Call your orphan (Blood Elf for horde, Draenei for alliance). Both are ADORABLE little girls. I wish I could just keep my orphan year-long.
The Outlands Children's Week quests are much much easier than the pre-60 ones. There's less running and flying due to Outlands being smaller, and the people doing the quests are more likely to have epic or flying mounts.
Alliance and Horde both have to go to the Dark Portal, which you probably came through to reach Outlands (if you didn't, it's the easternmost point of Hellfire Peninsula). Some have said they have trouble getting the quest complete here: make sure she's put away, get to the bottom of the stairs and take her out, then go up the stairs. She'll react in the middle of the staircase. If she doesn't, keep running around, put her back and take her out again, try different things. If that doesn't work, drop all the quests, destroy the whistle, and go get a new one.
Alliance goes to Auchindoun's Ring of Observance in Terrokar. Just fly or ride south of Shattrath until you hit the center of the burnt-out area.
Horde gets to go to Sporegarr, which is in the far western part of Zangarmarsh, to see her friend Hch'uu. (If you don't have flight paths here, there's one at Cenarion Refuge in the east and Zul'jin in the west.)
Speak to Hch'uu to finish the quest.
The Alliance orphan wants to visit her friend Jheel at Aeris Landing in Nagrand. Aeris Landing is the Consortium outpost where you pick up monthly jewel bags.
Next for Horde is the Throne of Elements in Nagrand, one of my favorite spots in the whole game for beauty. If you have the Garadar flight path, swim north across the lake and you'll hit the spot. She'll run up to the fire elemental and get zapped, which is adorable.
Alliance goes to Exodar and keep going down until you get to the basement. There's a Naaru there she wants to see.
Horde heads to Silvermoon City and down, just to the east of the entrance. There should be a bunch of taurens on a balcony. Bring out the little girl and she'll try to impress them.
Both factions get to Tanaris and head to the Caverns of Time (the mountains at far east, center). The events here are the same. Go see the dragon in the middle. Something interesting will happen. Then go get the quartermaster and buy your kid a toy dragon.
Return to Shattrath and turn the kid in for something not quite as adorable as your orphan:
[Sleepy Willy] Not as cute, but has great novelty value because he'll zap and kill little level 1 critturs (like rabbits or frogs). He's one of the very few interactive small pets.
[Elekk Training Collar] As my friend said, an "UNBEARABLY cute" miniature elekk named Peanut.
[Egbert's Egg] Gives you Egbert, a gawky, COMPLETELY SPASTIC half-hatched pink hawkstrider. He'll race around you in random directions.