By fast train, it takes about 1.5 hours to go from Zhangye West Train Station (Zhangye xi) to Jiayuguan South Train Station (Jiayuguan nan).
Jiayuguan is pronounced 'gee you gwan'.
Here are the instructions:
1) 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.
2) To go to Zhangye West Train Station, Bus 22 can be caught in the downtown area. It costs 2 CNY.
The train station is the last stop for Bus 22. The train station is not far from the city center so taxi's only charge about 10 to 20 CNY.
3) The fast train arrives at Jiayuguan South Train Station (Zhangye nan) which is about 5 kms from the city center. Taxi's to the center will cost around 20 to 25 CNY and they use a meter.