Hiroshima and Takayama are other good places
. Visiting Hiroshima was a good experience and seeing the Peace Museum is worth the trip itself.
Takayama is a really great rural type of town. Lots of scenery and places to go,wander and see (Takayama is known for it's carpentry and it has preserved alot of the traditional houses & streets)
You can stay in hostels for both places K's-house (Hiroshima) and J-hoppers (Takayama) so the accomodation will be cheap. Both places are accessible via Shinkansen (Bullet train). Although you may have to take an alternative standard train ride to go to Takayama (I am not 100% sure on this though)