I would recommend catching a bullet train (Shinkansen) at least once, although they can be quite expensive. I did get to see Mt Fuji but you may wonder why I do not have any section on my visit to Mt Fuji. Well the day I went I had booked a trip with www.viator.com a really worthwhile trip company to use on your first visits to Japan. Sadly although the trip was wonderful the weather was not! I never saw Fuji all day on the trip. Maybe next time.
This is Mt Fuji viewed from the train on the way back from Kyoto. Getting the train to Kyoto from Tokyo is very quick and only takes a few hours but the last time I travelled I got the over night bus and let me tell you if your on a budget it’s the way to go!