For the sake of putting in the first answer, you should surely rock on over to the
In China, Beijing is the statist, goverment city, and Shanghai is the groovy commercial city. (A bit like, say, Washington V. Los Angeles, or Canberra V. Melbourne, or Brussels V. Berlin.)
@MichaelSeifert has already mentioned the big one in Beijing, but I recommend the Shanghai Museum.
China's "museum boom" of the last decade...
China, as the ultimate nouveau riche entity on the planet, has gone through it's "museum boom" (you can read endless ... sarcastic ... articles on this by Westerners who don't like having it pointed out that museums are pretentious), so, your desire for cultural artifacts will be well catered-for in all major cities.
Simply click to wikipedia to get an idea of the absurdly large collection of Culture on hand at the Shanghai mega-Museum. It's the Alibaba.com of culture.