When you go to Vault City, enter the customs building, and talk to Wallace. Tell him that Skeev is involved in illegal practices, and show him the fake citizenship papers. Skeev will be dealt with, and you'll receive a free day pass. Enter the city afterwards.
Once you get in the city, go to the City Council, and talk to McClure. Tell him that you have repaired Gecko's Power Plant, and you'll be granted citizenship for Vault City. Head for the Vault City Vault now.
In the Vault, talk to the woman in the blue jumper. Tell her your name, and ask about Vault City. Tell her that you've noticed the similarity of citizens, and encourage her to explore the outside world. If you're male, you can go out with her for some experience and donate sperm to Vault City for some more experience. If you have high Intelligence, you can discuss the pregnancy cycle and tell her about radiation causing sterility and abnormalities among the citizens for even more experience. Save, and enter the second floor of the Vault.
Once you're there, go to the far right, and fix the rattling vent using the repair skill for some experience. Also, save the game, and attempt to lockpick all of the doors in the area.
The main objective is to open the locked door to the southwest (the door is next to the one whose power couplings are broken). That room has a Voice Module, which is essential to beating the game. Anyways, attempt to pick the door, and after you open it, save the game. I know for a fact that a lockpick skill of 70% or higher will work. Electronic Lockpicks will also help (they're at the Navarro base). Be sure to loot everything that you can find, however.
Go to the third floor, and to the far left. Use the science skill on the Learning Terminal for some easy experience. Afterwards, make your way towards the right, and use the computer that is talking. Optimize the Power Plant disc. Save, go to the corrections center outside of the Vault, and tell Stark that you'll scout out NCR for him. Now, go to Gecko.