I agree that 10 days in Shanghai is a rather long time if you're there for tourism. The suggestion to hire a driver and go to Suzhou or Hangzhou is a particularly good one -- you'll get a nice contrast to the ultra-modern parts of Shanghai. Also, though this is a matter of personal taste, I find Shanghainese cuisine a little to "subtle" (read as "bland") for my tastes. China is a very good country to get around by air -- frequent, inexpensive flights with excellent on-board service. I'd consider a several-day trip to one of the secondary cities.

With that said, I'd recommend a visit to People's Square which has a large park frequented by locals. It's also home to an interesting art museum, as well as the Grand National Theatre where you may be able to catch a good performance of real (as opposed to "for the tourists") Peking or other local opera, ballet, etc.