I live in the Dupont/Adams Morgan/U Street area of DC, and I love it. I think some of the best places in the City are away from the monuments in the central core. What hotel are you staying at?
Here are stuff in the Dupont Circle area. There's a lot of other great stuff downtown, and in Georgetown. But I'll limit my response to the Dupont area.
Museums/Attractions/Sights:
The Philips Collection -
http://www.phillipscollection.org/Woodrow Wilson House -
http://www.woodrowwilsonhouse.org/The Textile Musuem -
http://www.textilemuseum.org/Embassy Row - Massachusetts Avenue west of Dupont Circle
The Ghandi Statue - Massachusetts Avenue and 21st street (or 22nd)
The U Street Heritage Trail -
http://washingtonhistory.com/UStreet/The African-American Civil War memorial -
http://www.afroamcivilwar.org/Dupont Circle
Best areas for Restaurants/Shopping
Dupont: Connecticut Avenue north of Dupont circle
Dupont: P Street west of Dupont Circle
Adams Morgan: 18th Street between U and Colombia
Dupont/Logan: 14th Street from P to S Street
U Street: U Street from 18th to 13th Street
(You could also take a cab to M Street and Wisconsin Avenue in Georgetown, which is a great place to wander around. Or you could take a cab or metro to Woodley Park.)
Theaters:
The Studio Theater on 14th Street -
http://www.studiotheatre.org/The Source Theater on 14th Street -
http://www.sourcetheatre.com/Lincoln Theater on U Street-
http://www.thelincolntheatre.org/Visions Cinema Bistro on Florida Ave & 20th - Indie Films -
http://www.visionsdc.com/Restaurants:
Expensive:
Gabriel - P street near 21st street
BeDuCi - P street near 20th street
Tiramisu - P street near 20th street
Local 16 - U street just west of 16th Street
Cities - 18th Street in Adams Morgan
Perry's - Columbia Avenue in Adams Morgan (The waiters used to dress up in drag for Sunday brunch. I don't know if they still do.)
Mid Price
Meskerem - Ethiopian - 18th Street in Adams Morgan - Adams Morgan has the largest Ethiopian population outside of Africa. Try Ethiopian for a uniqe experience. (You eat with your hands.)
Fasika's - Ethiopian - Across the street from Meskerem
Pizzeria Paradiso - Gourmet Pizza - P street near 21st street
Raku - Funky Asian - Corner of Q street & 19th
Thaiphoon - Modern Thai - S Street between 20th and Connecticut
Kramerbooks & Afterwords - American - Connecticut Ave just north of Dupont Circle
Lauriol Plaza - Mexican - 18th Street near T street. (Always a ridiculously long wait. Too long, actually.)
Inexpensive
Ben's Chili Bowl - Burgers, fries, and chili, a U Street African American Legend -
http://www.benschilibowl.comWrap Works - Corner of Connecticut and Q
Chipotle - Connecticult between Q and R
Delicious Cookies:
Firehook Bakery - Q Street Between 19th and Connecticut
Hope you have fun!