I hope you find some reasonable ryokans for 6. They’re quite traditional, not too expensive, and you get to experience the baths, or onsen, and traditional breakfast.
The Shinkansens (I probably couldn’t teach you anything about trains!) will get you places easily. Pay attention: Japan is “cut” in several areas and each have their own railway systems.
Depending on when you wish to go, some areas are more or less suitable (the south is almost equatorial, the north can be really cold)