The Bomb Shelter
One thing you need to remember here in Vault City is that the people don't take too kindly to thieves, so don't try to steal from the buildings here.
You need a day pass to enter the vaunted Vault City, go to customs and tell them it's urgent that you see their leader, once you convince them by sucking up to the customs person, he'll give you the pass for free. Remember that this will only let you in during the day, and you gotta leave if night falls.
In order to find Gecko, talk to the First Citizen who will ask you to stop the ghoul plant. She can be found in the Council up north.
As for becoming a citizen, if you have a high intelligence, you can take the test in the Council or you need to fix the ghoul plant in Gecko to become one and talk to Mclure about your citizenship problem.
A woman outside the Vault City gates had her husband taken as a prisoner. Go to the detention building inside and talk to the guard there and convince him that the person has a rare disease and needs to be released at once.
A family to east in the slums is in need of income and they need a plow. The arms dealer to the west has a plow he doesn't need, go get it from him.
Thomas Moore is a sort of "evangelist" in Vault City, talk to him, and you'll learn he really works for the New California Republic. Try to convince him to give you a package which you will deliver to Mr. Bishop in New Reno.
The woman in the acquisitions building will only give you the kit the ghoul in Gecko needs if you give her a wrench and tools. Tools are easy to come by, but the wrench is a different story. There is an easy and hard way to do this. The easy way is to talk to the kid next to the tent in the southeast of the courtyard and learn that he has lost his doll. The doll is just north of the kid, right of the bar. Stick awhile and keep clicking on the kid once you return the doll and you'll learn where the wrench is hidden. Do you really wanna know the HARD way? I didn't think so. =)
Information about Vault 13 is stored in the main computer of this Vault. Become a citizen, and they'll let you in.
In the second floor of the Vault, there is a creaking vent shaft, fix it using your repair skills or with tools and you'll get some fuel cells for your car.
Go to the Information Center and convince the guy there that books are much better than computer text and he'll give you some cool books to help increase your stats.
The guy in the Corrections building wants you to scout the area around Gecko, which is really free xp, so don't pass it up. He'll also have other similarly simple tasks, so be sure to perform them.
If you're looking for the Hy-Mag(Hydroelectric Magnetosphere thing), go show Mclure the economic disk you got from the greedy ghoul in Gecko and tell him that you need to fix the reactor there, he'll tell you to go to Acquisitions and get the part there.