I spent just over a week in Japan, and did the following:- 3 days in Tokyo, then got a bus to Mt Fuji, and spent a couple of days there. Can only climb Fuji in the summer as all the infrastructure shuts down for winter. Stayed in a hotel with a jacuzzi over looking the lake and Mt Fuji, very relaxing.
Then got train to Kyoto. Fantastic place. Lots more to see in Kyoto than Osaka. Spent a few days in Kyoto seeing temples and wondering around. Some very good restaurants in Kyoto.
Then went on day trips to Hiroshima and Osaka from Kyoto.
Got bus / cab to Osaka from Kyoto to fly out.
As a foreigner you can but a Japanese train ticket for a few days / week, but you have to do this outside of the country. It seems pretty expensive, but when in Japan, and you realise that everything is nearby by train, you use them a lot more than planned ie Hiroshima is over 300miles from Kyoto, but only just over a couple of hours by train. Well worth it in the end.
The other thing is it is supposedly crazy busy in spring, as everyone is there for the cherry blossom. I was there in autumn and it was busy then as people are there to see the leaves changing colours.