There are Limited Express trains "Kaiji" and "Azusa" departing from Shinjuku or Tokyo station then reaching Kofu in one hour and half (about 3600 yen without reserved seat) and then on to Matsumoto. It is also possible, for about 2300 yen and an extra hour of travelling, to take local trains of the JR Chuo Line Rapid Service, switching at Takao.
The Chuo Line runs right through Kofu and accepts Suica cards.
Also in Kofu is the Minobu line which runs from Kofu, through Minobu, to Fuji in Shizuoka. This is a very quaint railway line, it is single tracked in most places. Getting into Kofu one must take a ticket from a dispenser on the train when you board. Going away from Kofu one must give their ticket to the driver as they leave.
The Minobu line serves as a local line in Kofu, having many stations within the city.
There are highway busses between Kofu and Shinjuku which on paper take two hours; this is shorter than the local train and challenges the express. In practice however traffic can delay this journey to over three hours. The advantage of the bus is that it is cheaper, costing about 2000 yen. The reservation can be done through [url=https://www.highwaybus.com/rs-web01-prd-rel/gp/info/lineDetail?lineGroupNo=1&lineId=120]this page[/url] (in Japanese)