Kōfu, Yamanashi (甲府市) is the capital of Yamanashi. It is the heart and soul of Yamanashi and it's central location means that many events and meetups take place in or near Kōfu. Many people that live outside of the city will refer to any urban area in the Kōfu Basin as "Kōfu."
When going from Kōfu to Shinjuku, you can take the Limited Express. Alternatively, for half the price (and at the cost of an extra 30 minutes) you can take the locals and switch lines at Takao.
If you are heading south, you'll need to take the Minobu Line. This line is infamous as one of the worst in Japan. It is single-tracked most of the way, which causes occasionally long waits for other trains to pass. It also has a tendency to hit deer and cause delays. However, it will take you down to gorgeous Shizuoka without a single transfer.
There are a number of bus lines running through Kōfu. There are local lines to take you to most places in the northern part of the prefecture, as well as buses to take you to Tokyo and other major cities. Highway Buses operate between Kofu Station and Shinjuku. This is a convenient and inexpensive way to travel, and you don't have to risk standing on the trains.
Notably, there doesn't seem to be a simple bus route to take you to the Minobu area. If you're heading that way, you will probably need to take the Minobu Train Line, and then a bus from one of the stations in town.
If you are headed to the mountains, there is a bus that will take you to Hirogawara in Minami Alps National Park. This is one of two ways to get to Hirogawara in order to climb mountains in the vicinity of Kitadake. The other option is taking a bus from Hayakawa. The park is not open to public vehicles, and is only open for 6 months out of the year.
Eating and Drinking
Kōfu is a hotspot for Yamanashi's rather limited nightlife. All kinds of ethnic foods, a variety of bars, izakaya, and even a few small clubs are present within the city lines. If you are close enough to stop by on a weeknight, you should have enough dining choices to keep you from going too crazy.
As for Kofu specialties, most of them are Yamanashi specialties. Keep an eye out for torimotsu, houtou (a delicious soup with thick noodles and usually pumpkin), and the occasional bit of horse meat (basashi is the name for horse meat prepared in the style of sashimi).
There is a Starbuck's on the south side of Kofu Station which is a frequent haunt for studious foreigners. In fact, most of the food and drink business will be found on the south side of the station, spreading out on either side from Heiwa Douri.
The north side isn't completely barren in this regard, however. There is a very good Korean restaurant just a few minutes walk north as well as a new coffee shop named Akito (look for the wooden-building about 3-5 minutes due north, it's hard to miss).
There is a popular Okinawan bar to the west-by-southwest of the station. Make reservations if it's the weekend, because this place gets packed.
There are several themed bars spread out across various floors of the taller buildings on the south side. Be aware that most of these will have a small seating charge, but it can be a nice place to relax and some have amazing music collections that you can request tunes from.
The local expat bar is called The Vault. If you need to see some foreign faces, that's the easiest place to find them.
Don't be shy when looking to grab a drink or bite to eat. Most people are very welcoming and flexible to any language barriers.
There was once a great hamburger shop called Hot Dog, but the owner has moved shop to Harajuku.
On the east side of the station is the Kōfu Castle Grounds. While there are few structures present and you'll have to imagine the "castle," it commands a beautiful view of the basin and is a lovely place to relax. It is definitely with stopping by! There are also picnic benches and open grass that you are free to use.
Yamanashi has a new library in Kōfu and it's fantastic. It's a great place to study (if all the seats aren't occupied by sleeping students) and has a really nice atmosphere. You don't need a library card just to check it out, although you're welcome to apply for one.
There is a small plaza on the north side of Kofu Station that has small festivals rather frequently. If you're in the area and need to kill time, swing by to see if anything is going on!
Kōfu has a soccer team and a bit of pride to show for it, so you've probably seen the blue and red colors around town. The team is named Ventforet Kofu. The games are pretty popular, so if you are interested try asking around and see if you know anyone going.
There is a tiny "zoo" in Kōfu. Please avoid visiting it, as it is nothing but depressing.
There are a number of boutiques, again focused mainly along the south side of the station. The best place for fun shopping is probably to the east on the south side of the station. Walk for a few minutes and keep your eyes open, you'll recognize when you hit the shopping district.
There are various hotels in Kofu, including the ever-present Toyoko Inn near Kofu Station.