Fallout/Vault 13

Examine the corpse to the right and take everything (24 10mm AP ammo and a knife). There are 20 rats in the Vault Cave; they are quite easy to kill. Kill them with a knife or brass knuckles (don't waste ammo). You don't have to kill all of them, but it will bring 500 experience. When finished, exit Vault 13.

Leaving the Vault
Leave the Vault entrance, and head towards Vault 15. After traveling for a day, choose to either continue to Vault 15 or return to Vault 13 for a number of sidequests. Returning to the vault is best done at 8:00 in the morning, saving the water thief quest for last. Most of the vault dwellers will refuse to talk to you during the quest.

Supplies
In the first floor locker (near the big vault door), there are some flares. To get some extra weapons, go to the command center (third floor) and repeatedly tell the overseer that you haven't found the water chip. When he says "Please... tell me that you have found the chip..." tell him that you'll need weapons to complete the quest. He'll give you some 10mm JHP ammo and two stimpacks. Next, go to the armory and talk to the guard. Ask him which people have access, then tell him "sounds like only people who have need can go in." Afterwards, tell him that you have need. He'll unlock the door for you, and you can get a shotgun and 40 shells alongside some other ammo. You can also go into the water storage room on the third floor to find a first aid kit, stimpacks, and some mentats.

In the library (westernmost room on the third floor), there are two learning terminals that can be used. One is near the entrance, to its right. The other is on the northern end of the library. Use the science skill on each of the terminals to gain a total of 700 experience (350 per terminal).

Quelling rebellion
On the second floor, to the north-west, find a red-haired vault dweller named Theresa. If you have high enough speech and intelligence skills, tell her the journey is going well, and she'll begin to say the Overseer is paranoid. Tell her the Overseer is trying to protect the Vault dwellers and you'll receive 750 experience (by telling her this, the rebel faction is calmed; an alternative method is to simply get into a rebel meeting and kill the rebels).

The water thief
Go to the south end of the second floor and speak with Lyle and Cindy. Lyle is directly south of Theresa, and Cindy is the redhead to the southeast. If they don't talk about the water thief, rest up until 8:00 A.M., then head to the third floor. At this point, the water guard should be interested in talking about the water thief. If none of them talk at this point, the 30 day mark may have to pass before they'll discuss the matter.

Once someone informs you that someone is stealing the vault's water, hide in a corner on the third floor and wait until midnight. A Vault dweller will come up in the elevator. Wait for him to enter the storage room. When he leaves, talk to him, then tell him you are looking for the water thief. Threaten him if he resists, then search him. You'll receive 1000 experience.

Off to Vault 15
Finally, head toward Vault 15 and stop in the city enclosed in a green circle on the map.