I lived in Seattle for 4 years - great city to visit and definitely not your typical tourist destination! Time of year has a big impact on what you do, as the weather can really limit/enhance your options.
If you're there anytime but winter, I would highly recommend a trip out the San Juan Islands - whale watching, sea kayaking, relaxed island living. You can take a ferry from Bellingham to San Juans, or better yet a sea plane from Lake Union in Seattle!! If you don't have time to get all the way out to the San Juans, at least take a ferry from city centre to Bainbridge Island - maybe consider staying overnight there.
If you're there in winter, there is decent skiing only an hour outside Seattle (Snoquamie Pass or Mount Stevens are close small ski resorts - they even have night skiing!). There are bigger resorts a bit further out - Crystal Mountain is one that comes to mind. Easy enough to rent ski/snowboard equipment at the mountains.
Seattle city centre is very small and gets really quiet after 6pm, so it's worth a visit, but not for more than a day. Pike Place Market is a must see. Pioneer Square is the place to go for nightlife. Try to catch a baseball game at the new stadium near Pioneer Square for a real slice of Americana. The Seattle Centre - home of the world's fair and iconic space needle are not too far from city centre and worth a half day to go up the space needle, visit the Experience Music Project (music museum created by Paul Allen), catch a Sonics basketball game (definitely an amazing atmosphere), and maybe the Pacific Science Center / IMAX movie if you were into that sort of thing.
A must do - take a boat ride from Lake Union (near downtown seattle) to Lake Washington - really great to see a working lake (complete with sea plane "runway" and to go past the exclusive waterfront property on Lake Washington (right past Bill Gates' pad)
For a taste of where the locals hang out, Check out the Greenlake area for a few hours - just south of the city - really great spot for a walk around the lake to see how the locals live. Also not too far from there, the campus of University of Washington is pretty gorgeous and worth a walk around.
If you are looking for guaranteed sunshine (assuming you're not going in winter), I would highly recommend looking into a wine tasting tour on the eastern side of washington state for a day. It is a complete contrast from the Seattle area - think semi-arid, complete with rolling tumbleweeds and cowboys! It's becoming quite the wine growing region because of the good weather. I love the heat so I'm a big fan of eastern washington, but some people prefer the greeness of Seattle.
Think that about covers it! Feel free to PM me if you want more details about anything. Delighted to hear of someone willing to make that HUGE journey all the way across the US!!! Seattle hardly ever gets Irish visitors compared to the rest of the country!