Head inside and take out the two Super Mutants in the lobby, go up the stairs (past the security glass) and through the doors into the makeshift Vault tour. At the end of the tour, head left into the doors to the West wing and you have two choices on how to proceed from here. If you take the route down the stairs to your right, you'll need to take out two Super Mutants and then head through the planetarium to the dish (which also has a Super Mutant Master waiting for you here). The left-hand route has a series of stairs and three Mutants to take care of (ultimately leading to the same place). So kill all the mutants, grab the dish and head back outside again. Take it to the Washington Monument, use the code you were given on the Terminal by the entrance and place the dish at the very top. Now fast travel back to Three Dog and have a chat to finish the mission!
Your next destination through the Wastelands is Rivet City (so fast travel here) and be sure to have a chat with Dr Li in the science lab. Once you've finished nattering to her, head to the Jefferson Memorial (which can be found south-west of Rivet City) and head in via the Gift Shop entrance. Clear out all the Super Mutants (being careful to watch out for the Turret in the first main room) and head into the Rotunda. Walk up the stairs, grab the Holotape off the keypad and once you listen to it, you'll be directed to Vault-112. So check your map and Fast Travel to the nearest place you have been to (closest to the marker), and when you get there you'll actually come across what appears to be just a simple garage, all on its own.
Once you're inside the garage, kill the Mole Rats and activate the electric switch in the main room. This'll open up a hidden stairwell which leads to Vault-112, where you'll be approached by a Robobrain so equip the Vault suit he gives you and head on downstairs. You'll come across a room with various pods in a circle, so head on down (grabbing any medikit gear from any adjoining rooms) and get inside the only Tranquillity Pod available to you.
You'll now appear in a sepia-toned world (supposedly a typical 1950's American neighbourhood) and you have two choices on how to complete this next area. You can either go for the good karma option (which takes less time), or the bad karma option (which takes longer but is much more interesting). We'll cover both options available to you regardless:
Good Karma Method: If you head to the Dither's house (found to the north-west of your starting point) and speak to 'Old Lady Dithers' regarding what's going on, she'll tell you that none of it is real and that you need to head to the abandoned house and activate the Failsafe terminal. Once in the - very dark and creepy - house, you need to activate the following items in the following order to give you access to the terminal: Broken Radio, Glass Pitcher, Garden Gnome, Glass Pitcher, Cinder Block, Garden Gnome and finally the Glass Bottle.
Once the terminal appears in front of you, select the 'Chinese Invasion' program, read the manual, activate it and head outside to see a bunch of Chinese Army men gun down the entire village for real! Once that's done you can speak to Betty in front of you to find out what's been going on, or you can head straight through the door beside you to finish the quest with positive karma.
Bad Karma Method: If you're feeling rather devilish, you can complete this mission a different way. Once the mission starts, head over to Betty and agree to make Timmy cry by any method you wish (punching him in the gob is the quickest way) and head back to Betty for your next task. She'll ask you to break up the Rockwell's marriage (without killing either of them), so head into their kitchen, pick up the rolling pin and bludgeon Martha Simpson to death with it. Take the pin back to Mr Rockwell and convince him his wife did it.
Now head back to Betty for your next task, which is that you've got to - creatively - kill Mabel Henderson. There are a few ways of doing this, but the quickest is to go into her kitchen and fiddle with the cooker's pilot light and then ask Martha to bake you a cake. The resulting fire will get you the results that you desire. Now head back to Betty for the grand finale...