As a native Tokyoite, I recommend: Akihabara, O-Daiba, Takeshita Street in Harajuku (to see the young people dressed up), the imperial palace, and the Tsukiji fish market (get there really early). At night, Roppongi, Shibuya, and Shinjuku are fun. Chinatown in Yokohama is not very far from Tokyo. Lots of tourists go to the Ginza but I don't see the attraction myself. Tokyo Tower is also probably a waste of our money, though others may disagree.
The best part about Japan is food! Make sure you are constantly eating -- ramen, soba, udon, sushi, sukiyaki, whatever. Go to McDonald's and get a Teriyaki McBurger.
Kyoto and Nara are probably the best places to visit if you're interested in cultural sites (Tokyo is lacking in this department). Hiroshima is also worth a visit.
I strongly recommend attending a local festival. You can google for "Matsuri," or check
http://www.japantimes.co.jp/festivals.htm, which is updated monthly. The Japan Times is also useful for finding other events while you're there.
If you ski, there is great skiing in the north, and although you don't need to go as far as Hokkaido, the further up you go the better the snow and the smaller the crowds. Try to find a place with natural hot springs ("onsen").
You've named what are probably the most interesting cities. I'd suggest looking up places like Osaka, Kobe, Kanazawa, Hakone, Kamakura, or Nagoya. If you know Japanese, you could also consider places far off the beaten path.