I've lived in DC since I was in the 8th grade.....
When I graduated from college, I lived in a couple of group houses for about 3 years -- I had the time of my life. Huge fun, and very cheap. It's a way to live in a nice house in a decent part of town, with a bunch of fun people who are around your age. Plus, you have an instant group of friends to hang with. If you don't know anyone who has space in their group house, check out the Washington Post, or perhaps craigslist.
There are also lots of people who rent out a basement apartment in their home -- something to keep an eye out for when looking through the Washington Post.
Currently, I live in Northwest DC with my wife and kids and dog. We live in a nice house in a quiet leafy neighborhood, where all the neighbors know each other, and where there are lots of nearby parks/woods, the Potomac River, and shops/restaurants. We don't have a nearby subway entrace but it doesn't both me at all -- I don't mind driving in and around DC; there are many, many biking opportunities in DC; plus, there's a metrobus line of you want to take public transportation.
It's true that traffic can back up at times -- but, hey, it does that in any city, big or small, during rush hour. Some people complain about their 1+ hour commutes but they live well-outside the city.
Regarding where to live, you really should just come here and spend some time over 3 days or so just checking out where you'll work, how long your commute will be, whether you want to be near a subway line or a bike path, or the hottest bars/restaurants, or within walking distance of the Verizon Ctr (hockey, basketball, etc.), etc. DC is very different from, say, New York, in that there are lots of residential areas with houses, yards, driveways, parks, and people walking their dogs down to the local pizza place. Some places though are: Adams Morgan, Georgetown, Dupont Circle, and Capitol Hill. Remember that DC has several universities, so some neighborhoods have lots of students/young people, such as Georgetown, Foggy Bottom (GW Univ.), American University Park, and the area around Catholic University, to name a few.
By the way, I always had my own car once I came home from college and I never regretted it. Unless you live right downtown in the business district, such that you'll need a parking space in a parking garage (nearly all apt buildings have one), I could always find street parking.