https://www.dol.wa.gov/driverslicense/moving.html

https://wa.gov/how-to-guides/become-resident

Hopefully those sites will get you started. Start by changing your address with the post office, then register to vote and get a driver’s license/state ID. No need to go to city hall, just the post office and then a branch of the Department of Licensing for the ID (if you have a car you need to register, you’ll also do that through the DOL). And you can print and mail in a voter registration form. I’d also recommend having your mail forwarded to your new address for a while; the post office will do this for you if you ask. Lastly, make sure you update your new address with your bank and/or any other financial institutions needed. Welcome to WA!