Shanghai has myriad neighborhoods, and it seems that something interesting is going on only a block or two away. The neighborhood in Jing An district, just north of the Jing An temple is a great local neighborhood, with schools, restaurants, and entertainment centers like karaoke halls and bowling alleys.
Gubei district, on the west of the city, has taken off as an area for foreign settlement. There are a few international schools out here, and a giant Carre Four supermarket. This attracted many great restaurants and pretty greenways around buildings to stroll around, a more laid back environment than some of the more downtown areas. This is one of the few parts of the city where you can see houses as opposed to massive apartment and condo complexes.
The area of Hongkou, north of The Bund and across the Suzhou Creek, is a great area for a stroll, especially along Sichuan Road North. The area is an interesting conglomeration of residence, manufacture, and shopping that is exemplary of everyday living in Shanghai.
Huangpu district includes more of the downtown areas, The Bund, much of Nanjing Road, the Xin Tian Di shopping and entertainment complex, People's Square with its abutting Grand Theater and museums, and most interestingly Yu Yuan Gardens. This is an area worth checking out during business hours, and after hours as the local community exercise in many of the greener areas is fun to watch. People practice Tai Chi, ballroom dancing, and other routines in the courtyard area in front of the Be For Time Teahouse just right of the main entrance to Yu Yuan Gardens.
If you want a short trip to give an impression of what much of the rest of China looks like, with additional access to some older buildings, it is worthwhile heading out on the subway to the end of the line 2 subway to the Zhang Jiang High Tech Area Station, and then catching a taxi or bus out to Chuan Sha. This town area on the fringe of Shanghai has some beautiful older buildings and pagodas, and gives an idea of what much of China looks like.
Hengshan Road and Huai Hai Road offer some great communities around the French concession area. Some of these neighborhoods are quite pretty with parks and trees everywhere, giving a quiet elegance. Foreign embassies can be found around Huai Hai Road as well as good window shopping and nightlife.
On the Pudong side of Shanghai, Lujiazui encompasses most of the financial district with its modern skyscrapers, an assortment of 5 star hotels, and a few notable attractions such as the Oriental Pearl Tower, Jinmao Tower (currently China's tallest building), and the still-under-construction Shanghai World Financial Center, set to be the world's tallest building by the time it's completed. In addition, you'll find some impressive museums such as the Ocean Aquarium and the world-class Shanghai Science and Technology Museum. Close by is sprawling Century Park, an oasis of greenery and lakes and a great place for bike riding, canoeing, or jogging.