Get a JR rail pass, which will make it easy and cheap to go almost wherever you want in Japan.
Grab a train map and go wherever you want along the shinkansen route. It doesn't matter, you'll find something interesting. The only place I would make 100% sure to visit and spend time in would be Kyoto, but other than that, Japan is so interesting you'll find lots of things to do no matter where you go.
Eat locally. Please please please please ignore the fast food. Just walk up to any corner food shop you find. Even if you have to gesture people will be nice and helpful and you'll get incredible food for very low prices.
Related to the last sentence, ignore the GB crew for traveling advice. Other than Drew as far as I can recall they barely explored the country and I always hear them talking about eating American fast food there. Not the place I'd go to for Japan advice.