I'd recommend spending time in either Tokyo or Osaka to get the proper Japan big city experience. Kyoto is also a fantastic place to go to if you want to check out shrines, temples and other things like that.
Depending on timing, Hiroshima is probably worth going to, if only to go to the A-bomb museum and memorial. If you do stay in Hiroshima, I'd recommend also going on a day trip to Miyajima island, containing one of the most famous Shinto shrines in Japan, over 9 years ago
We have been to Japan a few times. We spent 2 weeks in Tokyo on one trip. The different Tokyo city hot-spots are worth seeing including Akihabara (electric town), Asakusa for temples, Tokyo Disney and Disney Sea, Tsjuji fish market is huge and the department stores go on forever.
Make sure you stay in a traditional Ryokan or Minshiku for the traditional tatami mat and futon. Check google for dress and shoe rules!
Try a local bath house, ask for a "SENTO" and again check the rules.
Dont be shy with the electronic toilet but learn which is the "off" button first.
I would recommend going for a day trip to Hakone too. Its a great Japanese day out by train, walk (on the Edo walk), boat, cable car, Gondola and funicular railway. And all in the shodow of Fuji San.
Another day trip is the Nikko area with its cultural theme parks. Its harder to travel there as its more designed for locals.
So get your 14 day JR rail pass and get going! over 9 years ago