I would like to get a phone number of my own while I'm in China, that is a SIM. Can I just show up at a shop and buy it? Where should I go to get it?
You'll need to show your passport to purchase a SIM card. The law in China now requires "real-name" purchase of phone accounts. This is partly an effort to combat telephone frauds. Many Chinese now make payments through their phones, so thefts and frauds invovling phones have become a major concern. With the ID requirement, all phone numbers are associated with a real person, and authorities hope that this will help track down fraudulent perpetrators. Chinese citizens use their National Identification Cards. As a foreigner traveling in China, you should use passport to purchase your SIM card.
Major mobile communication operators in China include China Mobile, China Unicom, and China Telecom. There should be quite some of their stores in a given city.