Stug,
If you get a chance, Kyoto (the "spiritual capital") of Japan is an absolute must-see. I've lived in Japan for about two years now and if I ever get the opportunity, I will not hesitate to jump the "Shinkansen" (Bullet train) to Kyoto. If you're planning on making Tokyo your main stop, you're only about three hours by train to Kyoto and I *strongly* recommend it. It is a great chance to see the old-school side of Japan.
There are literally thousands of temples in Kyoto but I'll lay what I think (again -- I am not the authority) are the best ones.
Sanjusangendo
Sanj?sangen-d? - Wikipedia, the free encyclopedia
This temple is my personal favorite and is only a couple minutes taxi from the main Kyoto train station (where you'd head in on the bullet train). It's a massive temple filled with over a thousand Buddha statues where they used to have archery tournaments down the length of the temple. Also, they have 20 or so large wooden statues dedicated to Buddhist Gods that are of incredible detail (God of Wind is my fav -- he's a bad dude).
Kiyomizu-dera
Kiyomizu-dera - Wikipedia, the free encyclopedia
Giant temple on the side of a large hill on the outskirts of Kyoto with tremendous views. The temple is constructed entirely of wood and has a great walking tour. Don't forget to sample the sacred water!
Kinkaku-ji (Golden Pavilion)
Kinkaku-ji - Wikipedia, the free encyclopedia
Depending on the time of year, this might be the prettiest temple of them all in Japan. The rock gardens, ponds, and gold leaf-plated temple are really cool and the scenery is impeccable. Also, due to the walking area you never feel overwhelmed with people (a nice change of pace in Japan). Highly recommended.
Stug, these *barely* scratch the surface, I could probably go on for pages and pages concerning the cool places to see in Japan. Nara, Himegi, the Peace Park in Hiroshima, etc, etc. I really recommend doing the Tokyo thing (it is a great place) but also getting outside the mega-megalopolis. Osaka is another tremendous big city, very underrated by tourists, and, to me, it feels more "Japanese-y" than Tokyo does at times.
Don't shy away from anything though, and feel free to ask questions. I haven't done everything, but I've run the gamut from Tokyo Disneyland (hilarious) to traditional shows/plays.
Hope that helps!