Hi Arthur! Thank you for your question. Indeed people cannot seem to get enough of Japan and unless you stay for a long time - a really long time, you will not be able to see everything there is to see in the Land of the Rising Sun! Tokyo, Kamakura, Hakone, Mt. Fuji, Kyoto, Nara, Shirakawa-go are a few places that are very popular.
You may read this article also as reference.
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Wherever you go, you will find history, culture, food, nature, shopping and entertainment. This is a given. Tokyo will give you a much more bustling and exotic version of where you are from. Kyoto will impress with stereotypes of tradition come to life. Hokkaido will overwhelm with open nature. Hiroshima will envelope you with the solemnity of human decision making. Kyushu will surprise with its historical pride. Osaka will remind you that Japan is not only Tokyo and so on.
You received some good answers, but I’d like to add one more thing. Japan’s seasons and climates mean that Japan shows a different face every month.
New Years is a solemn holiday. The weather is cold and sunny in the Tokyo area. Plums blossom in February and cherries in March and April, followed by wisteria and azalea.
Summer in the Kanto and Kansai areas are sizzling hot, but Hokkaido has drier, cooler summers.
The autumn colors start in September from Hokkaido, and peak in Tokyo around late November and early December.
Then there is Okinawa, where you can go diving in a wet suit in December. The water is warmer than the air temperature.
When would you like to visit?
Japan is really vast and you will never really run out of options. But for recommendations, you may want to check out Tokyo by visiting these places: Akihabara, Asakusa, Ueno Park, Roponggi Hills, Shibuya & Shinjuku, Meiji Shrine and Yoyogi Park. Monkey Park in Jigokudani is also recommended. After Tokyo head to Kyoto and visit: Kyoto Imperial Palace, Kinkakuji and Fushimi Inari Shrine and head down to Nara to see the Todaiji Temple, Nara Park, and the Kofukuji Temple. Also, a trip to Japan ain't complete without seeing Mt. Fuji. So hop on a day trip to Mt. Fuji and might as well check the Fuji-Q Highland.