The main area for tourists is the Bund, which runs along the riverside in the centre of the city. It was once the Wall Street of Shanghai, where traders traded, and the people bought. Its home to some large buildings, and flocks of people wandering around. There are some good coffee bars and restaurants, but you shouldn't expect too much to see beyond the faded glamour of a bygone era.
Pudong is the most modern area of the city, and home to much of the mew building work as well as the airport. There is plenty of shopping, and several museums, as well as the Oriental Pearl TV tower to explore.
Puxi is the more Chinese area of the city, with winding back streets, and plenty of character. It's a safe enough area to wander through, and a good place to try out authentic local cuisine.
The Shanghai Museum is home to some 120,000 works of art, although around a third have never been seen. Amongst the treasures are remnants of the country's imperial past, as well as western works imported during Shanghai's golden age of trade. It's a real highlight of the city, and definitely not to be missed.