Shanghai Museum
The Shanghai Museum is one of mainland China's best collections of treasures. With four floors to explore, you can easily spend half a day or more learning about Chinese culture as you browse through bronzes, jades, calligraphy, and porcelain (to name a few). There is also a wonderful book/gift store and a good audio tour (inexpensive).
Walking Tours
There's no better way to see Shanghai than with a walking tour. Just exploring street life is interesting even if you don't have a destination in mind. Armed with a guidebook, you can enjoy a free stroll through Shanghai's historic neighborhoods and soak up some street culture along the way.
M50 Moganshan Road Contemporary Art Galleries
M50 is the name of the complex that has become the Moganshan Road Art District in Shanghai. Formerly a set of dilapidated warehouses, the complex, just south of Shanghai's Suzhou Creek, has been turned into the premier location for Shanghai's modern art movement. Take a nice stroll around and look at what China's contemporary artists are up to.
Shanghai's Parks
Most of Shanghai's parks don't charge an entry fee (some do, usually between 5-20rmb). Parks are a wonderful way to observe and even participate in Chinese culture. Folks head to parks early to exercise and walk. Kids play and grannies walk hand-in-hand while minding children. You'll usually encounter a group of elderly folk singing or dancing and you may even be asked to join. Bring a badminton set and hit the shuttlecock back and forth—you'll definitely attract some attention and probably a challenge match—but watch out, the locals are good!
Xintiandi
Xintiandi, also written Xin Tian Di, is an area of reclaimed 1920s-era shikumen¬ houses that were ubiquitous in the area until very recently. As recently as 2005, the entire neighborhood around the current Xintiandi was made up of these old lane houses that were home to hundreds of families. Most of them have been torn down, having the same fate as Beijing's. But Xintiandi stands out as an entertainment complex housed in rebuilt, remodeled lane houses of Shanghai. Be careful, while it's free to walk around, there is plenty to tempt the RMB out of your wallet.