If you want to have a truly "Japanese" experience head to Shibuya on a weekend night. There are some good restaurants and its interesting to see the sheer volume of people.

Ginza: really expensive.

Roppongi: have a drink at the top of Roppongi Hills Tower, which should give you a nice view of Tokyo. Roppongi also has a lot of trendy eateries, bars, etc.

Harajuku: its interesting in the day time. you will see lots of young people dressed up in really weird ways.

Shinjuku: anything you could ever want to do is here.

If you wanna have fun at night in Japan, try karaoke. It's quite different from the western version and most western visitors that i have taken love it.