What to see in Japan?

In late April, early May, I’ll be heading off to Japan for about ten days to visit an old friend. I’ll be flying into Tokyo, but I believe she lives in Nagoya. I’ll be a train ride away from Tokyo, but aside from that wonderful city, what should I see?
I know I’ll be arriving after the cherry blossom festival, and she’s already ruled out climbing Mt. Fuji (well, at least, if I go it will be by myself…she did it not too long ago and swore “Never Again!”).
So, anything big and touristy I should see? What about not-so touristy and cool but fairly unknown to silly little gaijin like myself? Just trying to get the most out of my first international visit in about 15 years. Thanks.

I’d like a Doper’s eye view of this.