I live very close to Roppongi, so I know the area well.
I suggest going to Mid-town to some of the nicer bars there. There is a new bar at Roppongi Crossing that gets pretty good action. I have chatted up a few chicks there recently and got some phone numbers.
Overall, I also generally...
Eliah,
That's the place. Friggin awesome burgers. I have tried most of the other popular places in Tokyo as well. As for me, any burger is a good burger --- but a Fellows burger is something special.
Toz