Washington DC is a must-see for any serious traveler, with North America’s largest collection of huge monuments and free museums, plus great nightlife and interesting neighborhoods.
This can be a tricky city for visitors since the downtown area is fairly small and hotels in the center can get away with charging a fortune. There are some cheap hotels in non-touristy parts of town, but honestly you are much better off in Virginia in the suburbs a bit, and public transportation does a great job of quickly getting you into the center during the day. On the other hand, cheap flights to Washington DC are usually easy to find due to 3 major airports in the region.
Almost every museum and interesting building is free to enter and tour, courtesy of the American taxpayer. This helps keeps costs reasonable, though food and drinks do also tend to be pricey in the main tourist centers.