Shinjuku, Shibuya and Harajuku are good for seeing the modern pop culture side of things. They also play host to a fair few niche shops which tend to sell (unintentionally) humorous items of clothing. Try to go on a Sunday to get the best out of it - that's when the kids are taking a break from homework to dress up all funky.
Iirc near Harajuku there are also a bunch of nice shinto shrine that you can view.
Roppongi is good for partying, as is Ginza - but Ginza tends to be more expensive.
If you like fish and sushi, go to the Tsukiji fish market (if it's still open?). It's better to go early because its less busy, but its worth going for the local atmosphere. You can buy things like tea and herbs there, and the sushi there is freshly made and surprisingly cheap.
I'd also recommend scouring the backstreets and alleyways whilst in Japan. That's usually where the best restaurants/cafes are hidden away.