Hi, Mr Gavin. I have just been back from Beijing and I should admit that it is really exhausting to see Beijing cultual attractions. You have four days and sugguest you choose some favorite places and only visit one everyday. I recommend Temple of Heaven, Forbidden City and Yong He Gong Lama Temple. 1 and a half days in chengde is enough.
Beijing - Chengde:
You can take trains and find train schedule here:
https://www.travelchinaguide.com/china-trains/
You can also get to Liuliqiao Bus Station of Beijing. There are regular buses to Chengde, 3 hours or so.

Chengde and Datong have no direct bus and train connection. You should get back to Beijing first and then take train (or bus if you prefer) to Datong. After arriving Datong, rent a taxi to Hanging Temple. Talk about the taxi fare in advance with the driver. 50RMB is enough. You will get to the temple half a hour later. Yungang Caves is on the other direction. I think 100RMB for the taxi fare from the temple to the caves is reasonable. Anway, rent a van to finish the one day tour in Datong. Wutai Shan and Pingyao can not be finished in one day, I think. Stay over night in Datong and get to Wutaishan in the morning. Get back to Taiyuan in the afternoon. You can choose to stay one night in Taiyuan, or go on to Pingyao for overnight there. Then take train from Pingyao to Xian. There is tourist city bus routes (306)to Terracotta Army on the east square of Xian Railway Station. You can rent a taxi of course. Pay 100RMB for single trip. Negotiate the taxi price with the taxi driver in advance. Are you individule backpacker？Not very easy to travel like this for a foreigner, I think. Better to plan the tour as much as possible. Any more questions, I would like to help. I am Xi'anese. :-)