Takeda Shrine stands where Takeda Shingen once ruled Kai Province from his fortified residence. The shrine itself is modern, but the site gives Kofu its strongest historical identity: moats, stonework, old trees, and a small treasure house around one of Japan's most famous Sengoku warlords.
Go early if you want quiet. The shrine is north of the station, so it pairs well with a taxi or bus ride before returning to the centre.