I was in Japan for 10 days in 2008. We stayed at a college in Nishiyomia (between Kobe and Osaka). I had a lot of time to visit Osaka and went to the Osaka Museum and Osaka Castle, as well as visiting Dotonbori (a famous shopping district in Osaka). I also went to Kyoto (another excellent destination if you want to see a lot of really old culture and shrines). Check out Nijo Castle and gardens, The Golden Pavilion and probably thousands of other shines, there's also Geisha Town and some great shopping and eating.
In Hiroshima and Miyajima Island (part of Hiroshima I believe) I would tell anyone interested in history to go and see the Peace Memorial Gardens and museum, the Atomic Bomb Dome etc. Miyajima is where the famous Torii is in the water. We also went to Himaji and visited the castle and gardens there, but didn't see much of the town except on the strip from the bullet train station to the castle.
I also took a bus to Tokushima Island to vistit Naruto City and see the Onaruto Bridge and the whirlpools there.
Pretty much anywhere in Japan is going to be awesome, but it is quite costly. We never got to go to Tokyo because of time constraints, plus the bullet train ride to Tokyo from Kobe would have cost $300 US dollars.
That said, I would probably tell you that if you want to go one place and stay there, then go to Tokyo and spend your time exploring the city. Traveling around is difficult without a guide or someone to let you know what to do and where to go in advance. The Japanese are very friendly and helpful to travelers, but the language barrier makes being lost very difficult, and maps? HA! All I can say about that is, "directions" do not work the same there as they do in America and finding a place by an address is um...I never could do it, and neither could my Japanese host on at least one occasion.
well, I guess I'm not very useful, but those are some suggestions of places you could visit at least. I hope your trip is a fun one!