You're both 19 right? So I would definitely recommend Harajuku during the day on the weekends (check out Yoyogi park for the skaters, bands, punks, goths, rockabillies etc) and Shibuya on a Fri/Saturday night. Check out Club Pure (if you like Hip Hop & R&B etc), it's a one fee (2500 or 3000yen) all you can drink night and there are heaps of young hotties. Roppongi has a lot of clubs (some dodgy) in a small area so you can easily go club-hopping. Like WarriorBuddha said, Akihabara is electronics and gaming central.
Here's an idea: go to
www.gaijinfinder.com and register (it's free) and say that you are looking for some cool friends (or chicks) to show you and your friend around the funky places in Tokyo. Definitely put up a photo (you get more hits). Chances are you might meet people with similar interests who would be more than willing to show you the sights. Better to know a local than to follow a guide book. Let me know how you go.