I was recently asked how long it took to get my personal number, which resulted in a super long reply that I figured could be posted for everyone else to see.
I very much recommend SFI. The best way to go is through Vuxenutbildningen. You have to register in person at Studievägledningen på Rosenlundsgatan 8 after you get your personal number. Studievägledningen under "Hitta till oss" you can find which trams and busses go there.
Have you gotten your residency permit yet? That's the first step. It took approximately 3 months for mine to go through. Apparently mine got approved a lot faster than most. The typical wait is over 6 months.
To get your personal number you have to register with Skattverket which is the tax agency: http://www.skatteverket.se. Presuming all the paperwork is correct, it takes between 2-4 weeks for the personal number to come in the mail. After that point, you pretty much have access to everything that Sweden has to offer!
Again, I whole-heartedly recommend SFI. Are you going to be looking for a job here or are you just coming for travel/fun? It helps to know someone who has some connections. Even with connections and work experience people seem to get turned down a lot based on their lack of Swedish. An even bigger recommendation would be to find a copy of Rosetta Stone and start on it ASAP. It will start you out with basic learning just as you would with a child. It kinda sucks at first, but we need to realize that it is building the foundation blocks of further learning a language. If it wasn't for Rosetta Stone I would not have been so successful in SFI. I know you're all probably packing and such, but I cannot stress enough how important it is that you need to know the language. As English speakers, we are not forced to communicate in Swedish because the Swedes are oh so nice and oh so eager to use their English skills, but their awesome English compromises our attempts at speaking Swedish.
Finding someone to practice with and forcing yourself to speak Swedish is really fun. I have a Russian cohort, and even though our Swedish is unbearably broken we try our best to communicate... in fact it's actually quite stimulating.
http://www.thelocal.se/followsweden/ is also a great site. I wish I had looked at these things before I came =p
Anyway, I hope that this information suits some people well!
-Jenna