Here's The Local newspaper guide for moving within Sweden.
Below is my own guide based on my experience.
Once you know your new address and when to move in, you are required to report your address change to Skatteverkets (the Swedish Tax Authority). This can be done easily online at www.adressandring.se although you probably need a Swedish speaker to help you. You essentially need to enter your personnummer on the first page, and then old and new addresses including the apartment number in 4 digits on the second page (if you don't know the 4-digit apartment number, enter the floor number instead). On the third page, you can apply for a mail forwarding service that forwards all your mail delivered to your old address in the next 12 months for 430 krona (195 krona for students). This mail-forwarding service starts from a week later. Therefore, you should apply by a week before the moving date.
You will then receive a confirmation letter within a week. The letter is enclosed with two forms. You have to sign both and send one to Skatteverkets and the other to Adressandring. Once you're done with all this, your new address will be reported to your bank and insurance company etc. So you do not need to visit your bank branch etc. to change your address.
If you need carton boxes to pack your stuff, go to Clas Ohlson at Drottninggatan 53. Or contact the following companies: Biltema, Rusta, Coop Byggmarknad, and Jula. They sell boxes for reasonable prices. The storage company Shugard (see this post) also sell boxes, but they are rather expensive.
Many Swedes appear to rent a small truck or a trailer to attach to their car at a Statoil gas station or at an OKQ8 gas station and drive it on your own for this purpose. (So if you ask around you may not be able to learn about good removal companies...) If you don't have a driving license, there seem to be several removal services in Stockholm. One such company is Stockholms Stadsbud. Call 08 640 5296. You can book a smallest truck and two men for around 900kr per hour (as of May 2009). They charge 50 percent extra for Saturday/Sunday services with a minimum of five hour charges. Their service is very professional and efficient.
If you find unwanted clothes in your wardrobe, there are several places to give away second-hand clothing. One of them is Myrorna at Östermalmsgatan 71. Another is Stadsmissionen.
If you want to donate your pieces of furniture and arrange the pick up at your apartment, call Myrorna at 020 140 0044 or Erikshjälpen at 08 368 820. Make sure you call them well before your moving date (at least 2-4 weeks before).