If after-school skiing is a priority, somewhere out I-90 would make a lot of sense. However, for skiing generally I wouldn't consider Snoqualmie Pass any great benefit. It's flat, short, and lower than about anywhere else. It rains there more than the other hills. There is one 2-chair at Alpental with actual terrain but it's really limited and the chair lines are near impossible. And it's not open at night. All the rest is good for beginners, intermediates are pushing it to find anything worthwhile. That said, night skiing is a different thing and it is fun there.
I live riding distance from St. Edwards Park at the north end of Lake Washington which has a number of cool things about it, there are some good, pretty easy XC trails, big enough for a solid hour and a half if you ride everything. Travel to Stevens which is vastly better than Snoqualmie is a little under 2 hours, Crystal is about 2 1/2 hours which skiiers like better than Stevens. I'm a boarder, too much pole-pushing and hiking for my tastes boarding but it's a killer mountain for sure, the hikes do get you to some unreal stuff.
"In my younger days" boarding I dreamed of going to Jackson Hole, for its famed steeps. I finally got there and had no problem with anything I found there, the double blacks aren't any steeper than stuff we have here at Baker, Stevens and Crystal. Whistler has some insane steep stuff. Snoqualmie is a kiddie park, apologies to anyone that loves it there.
If you're going to travel at all for biking, most of the travel is north, Whistler, Galby, and if it ever re-opens North Fork are all north. I-90 connects to I-405 pretty far south and makes those trips longer.
Airport from the north end of the lake is an easy half hour unless traffic is on, hour when it's clogged. Summertimes during construction can be longer if you catch it wrong. Airport from 90 is a little better because 90 connects south of Bellevue, and airport is south of that. I-405 also gets stuck south of Bellevue, but missing the big B would definitely be a plus.
All told I think Seattle proper doesn't give you skiing, not great for biking, no better than 405 for airport access. I'd focus on the 405/90 corridors.
I'll throw a wild idea out there - Woodinville. It's north but right next to I-405 and quick to the airport, on the way to Stevens, still short-ish to Snoqualmie, and Paradise is right there for biking. Quiet place if you like that. Being essentially right on 405 makes getting wherever easy too.
I would prioritize your goals and go from there. Short airport commute, live in Renton; easiest access to day/night skiing, North Bend; all night rave parties Belltown.

And, great advice above - rent first. That will let you soak in the place and figure out where YOU wanna live.