The Smithsonian museums are definitely all worth a visit, and they are FREE! It's also worth checking out the monuments - the Lincoln Memorial, the WWII Memorial, the Washington Monument, Vietnam Memorial, Jefferson Memorial, etc. You need tickets to go up into the Washington Monument (although it is currently closed), but you can otherwise freely and easily explore these.
You have to apply to visit the White House, and it's a pretty restricted tour. I wouldn't bother with that. The Capitol is free to visit and tour; you can order tickets or obtain them that day (time slots will be limited the longer you wait). You can also contact your particular senators or house representatives and they can arrange more private tours.
Arlington Cemetery is worth a visit too - lots of famous graves, but the Tomb of the Unknown is really moving to view.
There are lots of other great things to see and do - the Holocaust Museum, the Newseum, the National Zoo (free), Georgetown, the National Cathedral (in Georgetown area), the Library of Congress, the Supreme Court, etc.