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This is where the G.E.C.K. for Arroyo, your first major quest, is. There are two important things to get from Vault 13. The G.E.C.K. of course, from quest #1, and the NavCom parts for New San Francisco's quest #2, both in lockers on level 3 of the vault. Simply do quest #1 and you should have no trouble finding either if you check all the lockers on level 3.
There are at least two ways to learn the location of Vault 13. One costs you $10,000 in NCR and the other results from accessing a computer in Vault 15 (see that section). For the expensive way, travel to NCR, find Westin's estate (talk to Sheriff Dumont), give some booze to Saltbeef in the shed there who will then tell you about Vault 13. Next go find Doc Jubilee in NCR and ask to buy his painting of Elvis which he got from Saltbeef. Buy it, examine it, and you'll now have the location.
Before you enter the vault, make sure you have the Computer Voice Module, either from Vault City's vault on level 2 through a locked door, or slightly easier from Eldridge the arms dealer in New Reno's commercial row.
On level 3, Goris is the only one who can put the Military Base on your World Map, but for the MB, you can at least walk around until you find its green circle on your World Map... unlike Vault 13's location.
When you enter the Vault 13, talk to the Deathclaw named Gruthar and agree to help them for your first quest.
Quests & Opportunities
Here is a list of some of the quests and other opportunities in Vault 13, each with a brief synopsis containing at least one way to solve it. The remaining ways, and possibly other minor quests I will leave up to you.
1. Fix the Vault 13 computer:
As mentioned above, talk with Gruthar and agree to help them with the computer. You must have the computer voice module to complete this quest (again, see above). Once you have it, head to level 3 of the vault and use it on the terminal in the Overseer's room. After talking with Gruthar again, you'll get the G.E.C.K for Arroyo (woohoo!) and a fair share of xp. Now, if it's not too late, return to Arroyo with your G.E.C.K.!
2. Talk to Goris:
As you talk with Gruthar about various things, he should refer you to Goris to learn more about particular topics. Goris is on level 3 of the vault in robes. Talk to him and you'll complete this most difficult task. Goris can actually join your party as an NPC and place the Military Base on your World Map.