First stop for Seattle biking classes, resources, and general information is Cascade Bicycle Club.
For a Freeattle choice: check out the REI calendar! They regularly schedule FREE bike maintenance classes. Those classes fill up fast, so sign up early.
They might not be regularly scheduled, but check the event calendar of both Seattle Public Library and King County Library System sometimes offer FREE! bike commuting or maintenance classes. They are often tied to larger event themes like "urban self-reliance," Bike Month (May) or general community fitness.
UW's Experimental College sometimes offers bike maintenance or bike touring classes. While not free, they do have a discounted price for UW students.