I have lived and worked in the DC area for 3 years now, so I thought I'd trow my 2 cents in....
When I first came up I stayed with a family friend in Springfield, VA and I worked in Alexandria. Springfield = lots of traffic and lots of families.
After a couple months I moved to Clarendon (an area of Arlington, VA which is the county immediately west of DC, across the Potomac). I shared a large 2 bedroom, 1 bath apt for $850 ($1700 total) per month. By the time I moved out 2 years later the rent had gone up to 900-950 per person (luckily my rent never changed though ) Clarendon/Courthouse/Rosslyn/Ballston/Virginia Square are all nice parts of Arlington and are all very convenient to the city. You can hop on the Orange line and be at GWU within 10-15 minutes max. These are also nice neighborhoods if you want the conveniences of the city and the suburbs. All of the areas can be quite pricey though if you are looking to live in a highrise apt/condo building or if you want to be right on top of the metro. I lived in house that had been converted into 3 apts, we had driveway parking, and my landlord was awesome. I believe I was about a half mile from the Clarendon metro. If you're interested in Arlington, be sure to check out Craigslist, and respond to ads ASAP as it's a pretty popular area. You can get by without a car in these parts of Arlington, but I always had one.
Last summer I moved in to the city to the Capitol Hill/Eastern Market area. I now pay just over $1000/month to split a 2 bedroom, 1 bath in a rowhouse. The space is smaller than my place in Arlington, but we have a small fenced backyard/patio area. Street parking in DC can be a nightmare sometimes, but luckily my place is across from a small condo building w/ its own parking lot. I was a little worried about living in Capitol Hill at first - the surrounding areas are still undergoing the whole "gentrification" process - but now I absolutely love it. You can get by in Capitol Hill pretty easily without a car, but I keep mine so I can drive to my parents house, etc when I want to. (Also so I can drive to the grocery store when it's cold out ) If you're in Cap Hill you're using South Capitol, Eastern Market, or Potomac Ave metros. All of these are on the blue and orange lines which are very convenient.
If anyone has any questions about the area I'd be glad to try to help answer them