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Before you move

Apply for your residence/work permit at Migrationsverket (or confirm EU right of residence)
Join the municipal housing queue — it costs almost nothing and the clock starts now
Gather originals + certified translations: passport, diplomas, marriage/birth certificates
Line up a second-hand rental for the first 6–12 months (Blocket Bostad, Qasa, HomeQ)
Check what your current insurance covers abroad; plan Swedish home insurance (hemförsäkring)
Set aside a buffer of 2–3 months' costs — deposits and the personnummer wait eat cash

Your first week

Visit Skatteverket to register (folkbokföring) and apply for your personnummer
Get a prepaid SIM so you have a Swedish number from day one
Sort your transit card / app for the city you live in
Ask banks about a limited account that works while you wait for the personnummer
Walk your neighbourhood: nearest vårdcentral, grocery store, pharmacy, Systembolaget

Open a full bank account and get BankID — Sweden's digital key
Log in to 1177.se and register with a vårdcentral (health centre)
Sign home insurance (hemförsäkring) — landlords and common sense both require it
Join a union and an a-kassa (unemployment fund) if you're working
Sign up for SFI — free Swedish courses through your municipality
Accept every fika invitation. This is how Sweden opens.