Beijing is the largest city in China and the capital of China. If you can get past the pollution (which can vary by day, there is a tremendous amount of history and intrigue jammed into this metropolis.
Mutainyu is one of the two major Great Wall sites that are outside of the Beijing area. Phenomenal views of the wall along with a cement roller coaster track to take you back down again when you are done.
With Beijing being the hub of government, Shanghai is the hub of trade and finance. It is a more modern city with a mag-lev train into town from the airport and plenty of skyscrapers. Shanghai is a shopping and tourist mecca as well, but I never made it into those areas.
Shenzhen is an industrial center, everything you buy from Walmart is produced in Shenzhen. From iPhones to lawn chairs, it is all made here and then sent up the river in containers for export. It is amazing how fast it has grown as you can see by the 2 sets of pictures taken 10 years apart.