While I generally agree with Jessica about housing costs, the reality is that most new arrivals, particularly younger folks, share an apartment or a group house during their first year or two in the area.
It keeps costs down, and can actually help you to quickly build a network of friends.
Yes, check Craiglists. Just be aware that there are some scam artists on Craigslist. The legitimate folks will want you to come by to look you over (and vice versa), and to see what you would be sharing. Never pay any money up front, and you should be okay.
Another option is to speak to the hiring boss at your new place, asking if he/she has any suggestions or can put you in touch with those who might have something. At any offices, particularly with younger folks, there's a lot of turn-over and networking amongst the people.
By the way, 15-20 minutes from DC barely takes you to the Beltway--and most of the suburbs live well beyond the Beltway.
I agree with "greenfrogs" about Arlington. While not cheap, it is just across the river from DC, and it is quite safe. Lots of fun restaurants and shopping, and with access to Metro you don't even need a car. There are many young folks who share apartments and houses in Arlington.