Well, I'd say, since it'll be your first trip to Japan, stick with Tokyo, Kyoto, and their surrounding areas. Hokkaido, Kyushu, Shikoku, Okinawa etc. are obviously worth going to, but are quite out of the way--so basically, you'd have to plan a much longer trip or just pick one area to go to exclusively. (I assume you don't want to do the whole stop-photo-leave-say I've been there sort of thing. I also assume you're not rich.)
Tokyo and Kyoto are no less "authentically Japanese" than any other areas, and in fact there are plenty very nice things to see there. It really all depends on what kind of things you like. Museums, nature, landmarks, architecture, history, pop/fashion culture, food, clubs/bars...? I mean, why Japan? What specifically about it interests you?
In any case, I would also like to throw Hiroshima out there as a suggestion. I feel everyone should take a look at its Peace park and museum. It's a bit south, but easily reachable by Shinkansen (definitely DEFINITELY look into the JR pass if you haven't already, by the way). I took a night bus from Tokyo and it wasn't so bad, either. There is also a island nearby called Miyajima that I really liked.