I`m person who loves both modern and traditional things. For me Tokyo, Kyoto, Hakone, and Nara has their own interest thingy. Places in that cities I gave rank for the best 5 are Yanaka Town in Tokyo, Yasaka Shrine in Kyoto, Todaiji Temple in Nara, Silver grass in Sengokuhara and Lake Ashino in Hakone. Hope it helps you^^
Miyajima as I loved the shrine, the crazy deer and monkeys and being able to hike around the mountain. Kamakura as it was great to visit the shrines and temples and hike through the forest between them as not many people were on the hike so it was pretty peaceful. Hiroshima, as it is not a good place but somewhere everyone should visit just to realise how bad it was for the people who were living there at the time (I imagine Nagasaki is the same). Nara because it was another enjoyable day of watching crazy deer and enjoying everything in the park despite the crowds. Climbing Mt Fuji as it's the only time I've ever hiked at night on purpose and probably the most memorable sleep deprived sunrise I've ever seen. But there is a lot more to love than just these that I haven't even seen yet.