OP's criteria are still no definitively clear, but I agree with everything in jib's 1st post. He does say he wants to use public transport to get around... I think that makes Hokkaido not a great option. I love Yokohama personally, but Yokohama is very much like Tokyo in that main stations will be crowded and the trains will have lots of people. When one says he wants to see a country and stay off the beaten path, it's never clear how literally he means it. I have based myself in Kofu for numerous nights and have loved it because the town is laidback, the hotel has onsen, amazing views of Mt.Fuji, and shopping/food within walking distance, plus I know many great off-the-beaten-path spots around Kofu. However, I know Kofu won't work for most people, even those who say they want to be off the beaten path.
Anyways, I would recommend Okayama. It's a big city that offers plenty to do/eat, but it's certainly not Tokyo or even Kyoto. It is amazingly well-connected by trains to so many attractions that are both prime (Kyoto, Himeji Castle, Hiroshima Peace Park, Miyajima) and second-tier (Kurashiki, Onomichi, Matsue, Matsuyama / Dogo Onsen, Shodojima).