The typical Tokyo tourist areas like Shibuya, Harajuku, Shinjuku, Ginza, Ueno, Akihabara etc. are all perfectly fine. However, if you want to go slightly off the map (within Tokyo), I would suggest checking out some of the smaller, cooler places, such as Kichijoji, Shimokitazawa, and Takadanobaba. If you want a more chilled-out experience, away from the hustle and bustle of central Tokyo, I strongly recommend taking a trip on the Arakawa tram, which goes from Waseda to Minowabashi...it's a lovely old fashioned experience that will take you through some of the more 'local' parts of the city that are often overlooked:
As for rural trips, really there's so much to see and do in Japan, it's difficult to suggest one thing over another. My recommendation would be to pick one or two prefectures, then read about the culture, history, industry, and try to visit during local festivals etc.
Enjoy your trip to this beautiful country!