I love the wild places on the planet. And if you want that in Japan, there's one best place—Hokkaido. The best of those are in Hokkaido's national parks. When you're there, you can see some amazing scenery and then rest and be pampered in some amazing natural onsen, hot springs. This is a very different Japan than the big city (Tokyo or Kyoto). I love both Japans.
So, first go to Tokyo and Kyoto (and Nikko). Then, since you're already flying all the way to Japan, why not take a little longer and explore the wilder places of Japan? Sure, they may not be your stereotypical "Japan, Inc." but you would be surprised how much of the culture you can pick up in these wild places, too. The way to get to Hokkaido is by plane. Once you're there, the best way around is a bit of a dilemma. I like to relax on the trains, but they miss the best scenery. Busses are better for scenery, but the information is almost exclusively in Japanese (though a bit of experience in reading timetables, careful comparison of a bilingual map, and a little spoken Japanese makes it possible). A car makes for the best flexibility, but between operating on the left side of the road, trying to read the signs in Japanese, and either traffic or macho drivers, I find driving in Japan tiring.