We just got back from a short trip to visit friends on Anacortes Island, about 1 1/2 hours north of Seattle. We spent the first day in Seattle proper. I agree that Pike Place Market is definitely a fun place. You can spend a whole day there. Watch the salmon throwing, but don't overlook the quirky little shops "down under". And if you're a coffee addict like me you have to make the obligatory pilgrimage to the very first Starbucks store just down the street.
You might also like the Experience Music Project Jimi Hendrix Museum. I'm a big Hendrix fan, but even if you aren't, there's also a science fiction museum and hall of fame there which I'm sure you will find fascinating.
Being from Houston, I couldn't pass up the chance to go skiing at Steven's Pass. Great weather for it, but not sure how it will be in June.
We didn't do the Space Needle. Seattle has lots of tall buildings now so it just didn't look that impressive anymore, and as dmnemaine says it's too expensive.
edit: Oh one other interesting thing to do is to go see the Ballard (Chittenden) Locks. They have a guided tour of the locks and you can see the boats get processed through from one side to the other. They have an observation room with windows built into the side of the salmon ladder, and if you go during salmon spawning season you can watch the salmon swimming through.