So it's been a while sense this was last answered and some things have changed...
White House: Sorry, no more tours! You can thank the sequester for that. You can, however, get in if the President invites you, or you are doing official business. I live 30min away from DC and I have a friend who gets to go to the White House Christmas Dinner every year so it is possible to get in. You can also go when they have events, like Easter Monday on the White house Lawn.
Capitol Building: This one is easy to get in to see. Just go to the East side of the building to the visitors entrance, get in line, and buy a ticket! You'll get a time to report back for the tour, so I would buy them online so you can pick a good time for yourself. However, a better option is to go see your senator/representative in one of the office buildings next to the capitol. There you can actually meet someone important and you get a private tour of the capitol building by their intern (My high school has a class where you actually get an internship on the hill). You also get tickets to see the senate and the house, which you can only get via talking to your senator/representative. You get to ride the secret subway that takes people back and forth from the offices to the Capitol Building.
Also, while you are in the area, I'd recommend checking out the Library of Congress which is right across the street. You can get to it from the Capital Building via underground tunnel which is really cool (literally).
Also, if you have time, most senate hearings and almost all Supreme Court hearings are open to the public. Take a look on their respective websites to see what they are discussing that day. They can be really fun and who knows, you might even get into one of the medias pictures, securing your spot in history.