the best neighborhoods in DC to live in are Foggy Bottom, Dupont, and Adams Morgan. The universal consensus is to avoid NE, SW, SE if possible. Everyone says U Street is up and coming, but I have had a few friends get mugged over there. G-town itself is beautiful, but as you know the law school isnt in the neighborhood and the neighborhood requires a longer walk to the metro.
the law school is on the redline which makes dupont ideal if you want it to be easily accessible. Dupont is a younger neighborhood (young professionals, students, etc). Great restaurants, decent bars, etc. There is a little bit of a gay element to it, but not overwhelmingly so. Im just throwing that in if you have any hangups in that area. If you live in the northern part of foggy bottom you are closer to the farruget west (maybe north) whatever thats on the redline too.
These three neighborhoods are probably the best bet in the city. You can also look into silver spring. Its cheap and metro accessible.