Old Town Alexandria is cool as well and metro accessible with a little bit of a walk. I live near there and pretty much live right outside of it during the week as I stay at my g/f's place in VA. There is tons to do and the monuments are cool and you apprecaite them but there is a decent nightlife there as well.
Georgetown, Adams Morgan, Chinatown, etc...
You should go to a metro station that sells a smart trip as it would be better for you if you're going to be here a month. The paper tickets suck and ruin easily.