What is the best time for visit Beijing Xian & Shanghai.

Here is just my personal idea to my experience:
Beijing, March to May and around Sep. to Oct.
Xian, around April to May and Sep. to Nov. is suggested.
Shanghai, around April and Oct. to Nov. I have to remind you the densest rainfall is around May to Aep.

For the three destinations above, the hottest time is around July and August.