Xintiandi is a nice neighbourhood
http://en.wikipedia.org/wiki/Xintiandi
Walk along the Bund, see the wacky skyscrapers on the other side. Walk or take the tube under and go up the Jin Miao tower
http://en.wikipedia.org/wiki/Jin_Mao_Building
- great views but it's a smoggy city.
The Shanghai city museum is under one of these buildings and is very interesting.
You can visit old gardens like the Humble Administrator's garden.
We also took the fast train to Hangzhou for a couple of days, much quieter.
Peace Hotel on the Bund is a romantic old fashioned hotel to stay at (not that old fashioned but not a Holiday Inn)