DC is full of great restaurants. As mudbug said, Sweet Georgia Brown's is good. Have heard nothing but wonderful things about Bobby Van's, although I haven't eaten there.
I tend to eat at less expensive, often ethnic places.
A few favorites, both in the city and around are:
Amma Indian Vegetarian in Georgetown
Bombay Curry Company in Alexandria
Five Guys for burgers (chain in various locations, including Alexandria)
Lauriol Plaza for Tex-Mex in Dupont
Panjshir for Afghan in Falls Church
Pho 75 for pho in Arlington
Southside 815 in Alexandria for American/Southern
Taquiera el Pablano for super-cheap Mexican in Alexandria (v. small - can be crowded)
Pasha Cafe for great Egyptian food in North Arlington
I also like Ethiopian a lot - Adams Morgan has a few good restaurants (Meskerem has very nice atmosphere). Up and coming Ethiopian area around 12th & U with great restaurants like Dukem.
I also really like Levantes in Dupont - nice restuarant design by Adams & Demetriou with a great outdoor eating area when the weather's nice for mediterranean cuisine.
DC Coast (14th & K) is also nice.
Can't beat the Hotel Washington on 15th street for drinks with a sunset view.
If you are looking to grab a snack - Marvelous Market in Dupont & Upper Northwest (as well as other locations) has great snacks - especially their gingerbread - as well as sandwiches.
Fabulous Italian sandwiches (but not really good for sit down - just a few tables) at The Italian Store in North Arlington
You also may want to check out the best buys and very best lists produced by the Washingtonian magazine. Here's the link:
Never been to Raleigh/Durham, but if I were going, I would go to the Magnolia Grill - have a cookbook of theirs & it's supposed to be great. I would probably also check out Foster's Market there, too, although I don't know if it's really a restaurant as far as atmosphere goes.