By fast train, it takes about 3 hours to go from Xi'an to Lanzhou. Here are the instructions:
1) Take the Xi'an Subway Line 2 to the Xi'an North Train Station. It is the last stop going North.
2) Xi'an North ticket lines are very long. Advance tickets can be purchased online, for example on Trip.com (formerly Ctrip), but they still have to be picked up at the ticket counter. Allow 15 to 45 minutes for waiting in line.
3) The fast train arrives at the Lanzhou West Train Station (Lanzhouxi). There is a bus terminal on one side of the train station. Bus 31 (costs 1 CNY) goes to the West end of the Zhangye Road Pedestrian Street in the center of Lanzhou. Get off when the bus turns right one block after passing the big Mosque on the left. The bus takes about 30 minutes to get to the pedestrian street area depending on traffic.
4) To return to the Lanzhou West Train Station, Bus 31 can be caught on Zhongshan Rd, 50m south of the Pedestrian Street, on the West (left) side of the street, see attached map. The train station is the last stop for Bus 31 going west.