Hi Ms. Ramona,
I would suggest that you spend two to three days in Shanghai first, and then transfer to Beijng for 4 days sightseeing. Then since you are here for three weeks, i would suggest you take your time and see some other cities other than the ones you listed. You can visit some of the famous tourist' cities like Beijing, Xi'an, Datong, Nanking, Guilin, etc.
As for hotels, you can stay at a chain budget hotel if you want to save money.
Chris