You can try the one-two punch of dumplings:
- Yang's Fried Dumplings
- Jia Jia Tang Bao
They are across the street from each other near People's Square. Lines snake out of these small restaurants, but they are really yummy if you want to try authentic Shanghai style soup dumplings in two different styles, and they are cheap by American standards. Probably quicker to order take-out.
If you want to try more local food, you can try Jesse Restaurant.
I like all of CityRiver's suggestions. The river cruise at night is really nice. Go for a walk along the Bund. Xintiandi is sort of a westernized shopping/eating area. If you like tall buildings, you can also go to the Shanghai World Financial Center. It's the tallest building in Shanghai. There is a somewhat expensive observation deck or you can go a few floors lower and try to grab a snack with a window view at one of the Park Hyatt restaurants, also similarly expensive but then you get some food with the view.
If you want to see Chinese art, the Shanghai Museum is a really good place to go. It's free I think, and you can go inside just to sit in a quiet air conditioned place too.
Another popular shopping area is Nanjing Lu for several blocks starting from the Bund and going eastward.