Here in the US, we would normally... register the new address with the postal service (online), update the address on our driver's license and car registration (online if same state, in-person if new state), send an email to employer with new address details.
So as long as it's a move with a state, it's not too bad. Name changes (marriage, etc), gender changes, out of state moves, and other "edge" cases can be quite a bit more painful.
And of course, updating addresses with utilities/ISP and other non-governmental entities varies quite a bit.