There are a number of lovely Seattle neighborhoods that I personally like; Queen Anne, Freemont (Adobe, Google, Tableau and the center of the universe), U-District (University of Washington), Mount Baker (easiest access to I-90 and the mountains in Seattle), Montlake, Ravenna, Ballard, Wallingford, Green Lake, Phinney Ridge, Laurelhurst (Bill Gates Sr.), Madison Park (Tom Skerrit), The Seattle Highlands, and South Lake Union (Amazon).
All of these neighborhoods have a different vibe than each other and you could find one that appeals to you but unless you earn a fuck ton of money you will not be able to afford to rent a house with a yard. Also, access to the mountains is more difficult than the east side (of Lake Washington), but there are a number of climbing gyms throughout the city and some nice parks for trail running. Traffic is as bad as the media makes it out to be.
Renting in Bellingham is tough because it is a college town, but certainly less expensive than any of the Seattle neighborhoods. If you can work anywhere, somewhere like Preston, Fall City, Carnation, North Bend (Twin Peaks fame), Wenatchee, or one of the smaller towns near Bellingham like Ferndale or Mount Vernon would offer better affordability and great access to the mountains while being plenty close to climbing clubs and guide services.
I live in Issaquah, which in addition to being an easy 20 minute bus ride into Seattle is a great location for all outdoor recreation: world class trail running from my front door (the Issaquah Alps), 20 minutes to rock climbing in North Bend (Exit 32 and Exit 38), and 40 minutes to Snoqualmie Pass for lift serviced skiing, back country skiing, ski mountaineering, alpine rock climbing, and very accessible winter mixed alpine climbing due to the proximity to the ski resorts. It is, however, very much the 'burbs. Think Mercedes, Lexus, and BMW SUVs, families with young children, good schools, and no singles scene to speak of. It is more affordable than Seattle or Bellevue, but not inexpensive either.
If I were not tied to a job in Seattle I personally would look hard at the Mazama/Twisp/Winthrop, Leavenworth/Peshashtin/Cashmere/Wenatchee, and Bellingham/Ferndale/Mount Vernon/Anacortes metropolitan areas.