Seattle-Tacoma International Airport: Take Sound Transit Link 1 Line from the airport station to Westlake for downtown, Capitol Hill, or University District. It is the correct default: roughly 35–45 minutes to Westlake and about $3–$3.50. A taxi/rideshare is usually only worth it for heavy luggage, a late arrival, or a hotel far from Link; downtown rides commonly run about $50–$70 before surge and tip.
King Street Station: Amtrak Cascades, Coast Starlight and Empire Builder arrive beside International District/Chinatown station. Link 1 Line gets you to Pioneer Square in one stop, Westlake in two, or Capitol Hill in about 10 minutes; buses and rideshare fill the gap for hilltop hotels. Downtown is walkable from here only if you pack lightly—the climb is real.
Paine Field / Snohomish County Airport, Everett: Use it only if its limited flights are dramatically cheaper or more convenient. It is roughly 35 miles north of downtown and has no clean airport-to-Seattle rail transfer; a rental car or rideshare is the practical choice, usually 45–90 minutes depending on traffic. SEA is substantially easier for a Seattle stay.