First get the quest by gaining access to Little Lamplight, through the use of Slaver quests, the main stroyline, or the Child at Heart perk.
Right past the gate, you should see several children huddled around one that's wearing a party hat. Speak to Sticky, the birthday boy. When you do, he will state that he has just turned 16, forcing him out of Little Lamplight and into Big Town. Being a "grownup", he states that he can take care of himself on his trip to Big Town, though he'll take you offer to escort him there quite quickly.
Once you have agreed to help him, simply direct him to Big Town, which is a slightly lengthy walk due north of Vault 101. However, Sticky is one annoying bastard, barking every few in-game minutes. He is so annoying, even the Lone Wanderer has several options dealing with his quieting his overly-active mouth.
Walk across the bridge to Big Town and you have completed the quest! No rewards, though...
You can obtain Sticky's party hat by using the "What's with the party hat?" dialogue option when he is in your company.
The mines in Big Town that are laid if you use that option to help the place with the Explosives option think Sticky is an enemy. If you don't tell him to wait before nearing the bridge, he will run into the mines and blow up, though it doesn't really matter as there is no real reward here other than the location of Big Town and putting an extra person there.