What is the best way to travel Beijimg, Xian and Shanghai?

We would like to visit Beijing, Xian and Shanghai. We will be coming from Seoul and only have 5 to 6 days. What is the best way to route ourselves? Beijing to Xian and then to Shanghai. Can we go by train to Xian from Beijing then go from Xian to Shanghai? Any suggestions would be appreciated? Is there a route that we should book that does all three cities. Thank you.