You should do some online research about the John Wayne trail; last I heard, the 2.3 mile railroad tunnel at Snoqualmie Pass was closed due I think to rock fall/slide (pity, that was fun ...). I'm not sure that's still true, but do check.

My wife and I have ridden bikes from our house in the Puget Sound area up and over Snoqualmie Pass on that trail --- staying on trail as much as possible but certainly using some roads and bike trails. There are some navigation challenges to do a sort of "road minimal" approach, so again, look online and get appropriate maps to make that as painless as possible.

Even though you're hiking, you might look online at bike routes, and in particular bike routes that stay off major roads, as these might be nice options for stretches of your trip.

Best of luck!