Well, the short answer is: lots. Shanghai is a big city, and plenty of tastes are accounted for here. However, there's a difference between the more tourist-y activities you do if you're visiting Shanghai for a short time, and the kind of things you do if you live here.
If you are newly-arrived in the city or are looking for something to do with visiting friends, try taking the Double-decker bus around town. It takes in a majority of Shanghai sights and gives newcomers a good overall feel of the city.
Main article: Tourist activities
Main article: Expat activities
Cool areas to explore
Main article: Cool places to explore