As a person who moves around quite a lot, I can sympathise with your predicament. Not having a huge network of 'friends friends' as opposed to acquaintances didn't bother me so much when I knew I wasn't going to stay in a city for more than an academic year. But when I moved to Madrid, I wanted to put down roots a little and that meant looking to cultivate real friendships. And so I turned to the internet (because obviously THAT'S where you meet 'real' friends - ha ha!).

I learned about meetup.com through a friend who was living in the Middle East. It's THE place to find social groups. There's something for pretty much everyone. I also joined internations.org. This site attracts a slightly older, more business-y crowd, which is reflected in some of the activities on offer (and the prices for said activities). 

 A quick check suggests that both Meet Up and Internations have a reasonable presence in Seattle, so they might be worth a look.