If you have a smartphone, do yourself a favor and download One Bus Away. It is absolutely the best app for getting around Seattle.
From the Airport
Public transportation is available from the airport to the U-District. First, take the Central Link light rail from the airport to the downtown Seattle, Westlake Station stop. Trains leave approximately every 15 minutes and the trip takes 50 minutes. Once downtown, transfer to a metro city bus number 72, 73, or 74. Any of these three bus lines will take you from Downtown to the U-District.
Sound Transit & Central Link light rail information
Seattle Metro buses run throughout the Seattle area. Please see the King County Metro Website for fare and route information.
Yellow Cab of Seattle
Seattle Orange Cab