For your residence registration, you will need your passport and all documents proving your residency status. Note that your registration is tied to a physical address, so you will have to change it should you move. In that case, you will need to register your move with the local PSB before the relocation date, and then register again at the PSB of the location you are moving to. Failure to register with the PSB can result in deportation.
It is also advisable to register with your consulate or embassy. In case of emergency situations (i.e. natural disasters or political problems), they will have you in their database and will be best equipped to help you.