Shinjuku, Shibuya, imperial palace, akhahibara (sp) for electronics heaven, old Tokyo, and checking out their arcades for latest cool games. These are all the places I can list on the top of my head for Tokyo. Enjoy!
Good daytrip from Tokyo - Kamakura (well worth it, lots of interesting temples, big outdoor Buddha, good shops). Tokyo is short on "must sees" but totally amazing to see the spectacle. The Shinjuku subway station at rush hour is simply incredible. The Imperial Palace gardens are a nice green lung in the middle of the concrete jungle and worth a look.