When moving, you must make a transfer notification to the Swedish Tax Agency (www.skatteverket.se/flytta). When you register the move you will be registered in the new address. It is free and can be done in the Swedish Tax Agency's e-service or via a paper form.
Make your registration before you move or no later than one week after you move. Then your new address applies from the day you move in. The right address in the population register means, among other things, that authorities and many other actors in society have your real address. If you wish to forward your mail, you can order it at Swedish address change (www.adressandring.se).