Do I need to register with the police?
The regulations are subject to change, but now it is important for all foreigners entering China to register with their local police station within 24 to 48 hours upon arriving in Shanghai to fill out the Registration Form of Temporary Residence.
If you move in Shanghai, you are also supposed to re-register with the police. If you are not formally on the landlord’s contract, you should go to the police with a copy of the apartment contract, and a note stating that you are living there.
Do I need to register with the US Embassy?
The State Department recommends that all U.S. citizens living or traveling abroad register through their online system, so that in case of an emergency, they know how to reach you.
http://shanghai.usembassy-china.org.cn/registration_for_americans.html