There's only one rail line, the long, isolated Lhasa Express. In Qinghai the train stops at Xining and Golmud.
To go to most places within Qinghai you can use the province's extensive bus service. The hub is definitely Xining. From here it's possible to catch buses to most places in the province.
Xining has extensive buses connecting all parts of the city. The city buses usually cost Y1 per person. Taxi's start at Y6 and are an additional Y1.3 after the first 3km. After 9PM taxi's are Y1.5 per km after the first 3 km.
