Hey man...welcome to DC!
The best thing about DC is lots of stuff is free.
I'd spend a day at the National Zoo, on Connecticut Avenue...free.
Also I'd stroll around the Mall, and check out any of the Smithsonian museums that light you up...American History, Natural History, Air & Space, National Gallery....all free.
If you have a car and a nice day, Great Falls is excellent. In MD, do the Billy Goat Trail. 3 parts...from insane to moderate to EZ. Small fee.
Haines Point is cool too. Especially if you like buying crack.
Church lover? National Cathedral will make you feel like it's Christmas.
Old Town Alexandria is excellent...very cool and fun.
Taking a run? Rock Creek Park.
Biking? The Capital Crescent Trail.
But the best of the best is Georgetown. Night or Day, G-town is where it's at. Stroll along the Potomac, shop at the best places on Earth, then drink at Nathans, and Billy Martin's. And Clyde's if you want a nice dinner too.
If you're killin' it financially, do the Capital Grille, or DC Coast or Kinkead's for dinner.
If you're gettin' killed finacially, go instead to the Tune Inn on Capitol Hill, or Whitlow's in Clarendon.
Also...no matter where you're from, you can call your Congressional Representative and get hooked up for a tour of the Capitol, and even buy a US flag that flew above the Capitol.
Oh, and if you have time for headphone fun, Sound Works in Kensington keeps the whole Grado line!
Have fun, dude!