I will be in Zhangye and would like to go to Alashan Youqui to see the the desert.

As I know, there is no direct bus to Ejina. You can catch a shuttle bus to Badain Jaran first. Upon arrival, you may need to charter a car to Ejina to visit the desert, as well as the Khara Khoto and the forest. As the scenic spot is in the desert, you may not catch a taxi there, but you can take the special off-road vehicle to visit the scenic spot. There is no bus operating between Ejina and Dunhuang. You may also need to charter a car to your destination.