There is lots of information out there on tourist destinations in Shanghai, but why not get off the beaten path and see how the Shanghainese live?  Here are things you can do to experience Shanghai like a local.  
Go to the Park.
The parks in Shanghai are the hottest people watching spot in town.  Elderly Chinese people and their grand children flock to the local park to get down.  You can find karaoke, dancing, aerobics classes, tai chi, Chinese chess, martial arts classes, paddle boating, and the occasional shoving match.  Parks are integral to city life in China amongst the elderly.  Go out and bring a phrase book. People are very friendly and curious.
I recommend Zhong Shan park on Changning lu, on subway line 2, 3, and 4.  (Sidebar: lu translates to street).
Marriage Market.
On the weekends, visit the marriage market in People's Square.  The marriage market is a giant bizarre of parents seeking marriage partners for their sons and daughters.  Parents will bring their children's requirements and vital stats to see how well they match up with others.  Important marriage information include property ownership, education, hometown, income, age, and a bunch of other information that westerners feel is too invasive.  
Call to action:  see if you can find yourself a local husband or wife.  Don't forget to bargain!Shanghai marriage market
Eat some Xiao Long Bao.
Xiao Long Bao are a Shanghainese dumpling/dim sum treat.  Somehow, the Chefs get soup inside the dumpling, and get it to stay there.  Usually it is scalding hot, so watch your mouth.  
Good xiao long bao can be had along Dingxi lu by Yannan lu. 
Check out Qipu lu market.
The Qipu lu market is a whole sale/retail mall.  It is packed to the rafters with crappy textiles and apparel.  The complex is a bargain hunters paradise, although lately it is under a lot of pressure from the internet and local officials.  Just remember to bargain hard, as shop keepers will quote astronomical prices to obvious foreigners.  I suggest starting at 1/4 the original price. 
Check out Club 88.
Club 88 is the most successful club franchise in second teir (ie smaller) cities in China.  Come out and party like the locals.  Inside the club is complete sensory overload.  The decor is steam punk, the two hundred strong wait-staff look like anime characters, the music is ear splitting, and the place is so packed you can't move.  It goes way above and beyond a western club in terms of bombarding your nervous system.  Watch out for the beggars and the monkey men out front.    
 club 88
Visit a wet market.
Wet markets are still the preferred outlet for produce and meat in China.  In a country with little historical refrigration, freshness at market is a huge deal.  A wet market is packed with fresh produce, fruit, live ducks, chicken, turtles, fish, and extensive household goods.  
Blind Massage.
China has a long tradition of accupressure massages.  These were usually preformed by blind men, as one of the few jobs they could do.  These days there is lots of competition but many of the best massueses are still older blind men.  Most massage parlors won't have signs in English, but a few will say "Blind Massage".  You change into some pajamas and go into a room, often sharing it with other patrons.  Then a man with shovels for hands uses you as his squeeze toy.  Accupressure focuses on the spinal column and nerves.  It may not feel good during, but afterwards is heaven.  You should pay $US 10 - 20.
Visit the hidden Jesuit Library
Bibliotheque Ziccawei is a hidden gem.  Its a two hundred year old library run by the Jesuit church in Xu Jian Hui.  By some miracle, it survived the Cultural Revolution.  The library is still open for tours today.  Its quite an amazing place.

Shanghai marriage market
club 88