Go check out the bar at the top of the old Shanghai Race Club (the old 1930s vintage clock tower building) The bar provides great views of the park and surrounding architecture and is a relaxing place to unwind after a day at the Shanghai Museum. This place has a great history back to the British colonial era when the current park was a horse track.
Not for the claustrophobic but I suggest visiting Shanghai Museum (20RMB entrance is a bargain) to see Chinese calligraphy, furniture, pottery, jade ornaments. Wander past the Grand Theatre, the Peoples' Government, Exhibition Hall and marvel at other surrounding buildings.
Visit the underground shopping mall and especially the food court. ...
People's Square is an awesome area to visit while in Shanghai. Great shopping area underneath the metro. Fun atmosphere and has so much to do there. A great place to go and see how the people of Shanghai are.
Very fun to people-watch in good weather, but otherwise another undistinguished public square in China. Like most, it's crowded and noisy, so if you don't like crowds, avoid this place.