10 Things to do in Japan?

I will be spending 10 beautiful weeks in Japan from June-August! That means I have 10 weekends to see the sites! What are the top 10 places/things to see in Japan? I know I want to visit Osaka, Kyoto and Hiroshima for sure--but what sites? I will be living in Tokyo and I do not have too much money, but I am willing to do some overnights (youth hostels-YAY!). I am interested in cultural history above all else--I could care less about shopping. I am also not allowed to fly. If you could also provide me with information on how to get to the different sites from Minato-ku, Tokyo I would really appreciate it. If you would like, a basic itinerary for your suggestions would be AMAZING! Arigatou Gozaimasu!!