The west coast of the Izu Peninsula (about two hours southwest of Tokyo) is great for a taste of rural Japan. It's my favorite place to get away from the big cities. Great food, hot springs and no tourists (as long as you stay away from the east side of the peninsula). It's also easily accessible from Tokyo. Mt. Fuji is also nearby and worth a visit.
Things to do in Tokyo...
Yoyogi park on a Sunday, Inokashira park, Ueno Park, Sumo wrestling in Ryogoku, Fish market in Tsukiji, market streets between Ueno and Okachimamchi Stations (AKA ameyokocho).
One of my favorite things about Japan is
jazz kissas, but only if you like jazz of course.