I'm in Shanghai and tried to buy a sim card for internet using.

In SH, there are a lot of branch offices of China Mobile and Unicom. I know they sell the card to foreigners; you need to take your passports to register.

Well, for Wifi, most restaurants, hotels and malls in that city provides free wifi service. Usually there are signs of Wifi on the gate or wall, if you find such sign, that means you can have access to the wifi service.