I have 3 full days in Beijing. I have some areas I want to visit. Can you suggest the best route?

Mutianyu will take one day. The second day, you can visit Yonghe Gong (Subway Line 5, Yonghegong Station), Gong Qin Wang Fu (near Hehua Shi Chang, Shichahai area). Yuan Chonghuan's Tomb is in Guangqumen area. Guangqumen Nei Dajie's north, there is Donghua Shi Nanli. It is surrounded by residential buildings, next to Guangqumen Middle School, and behind a Vanguard Supermarket. Shanghaiguan is in Qinhuangdao City. It takes 2.5 hours by highspeed train from Beijing Railway Station to Shanghaiguan Station. Last hight speed train back to Beijing from Shanhaiguan at 17:39. Suggest you buy the return ticket in advance.