Everybody knows about the Burke-Gilman. Here is the wikipedia entry for the Burke. Here's a link about the trail's history from the Seattle website. It's a nice ride, but watch out for pedestrians and skaters. There is a missing link in Ballard between Fred Myers and Market Street. (They need to just get over it and build the trail – it can't be any worse than the Duwamish Trail by the glass plants on West Marginal Way. ) Eventually it is supposed to go on to Shilshole and Golden Gardens.
Here's the Seattle page describing current plans.
Watch out for speeding / stopsign tickets in Lake Forest Park.
Leave the Burke at 20th for the Ravenna shortcut .
Leave the Burke at the Freemont Bridge for the waterfront via either the South Ship Canal Trail or Dexter.
Leave the Burke at Huskey Stadium for the Arboretum bypass and the South Lake Washington Loop.
Leave the Burke at Juanita Way for the North Lake Washington Loop.
Leave the Burke (or maybe the Sammamish River Trail) at I-405 for UW Bothell, Canyon Park, North Creek and the Interurban.
Here is the Burke Gilman page from the king county web site.
Here is a 2013 plan for redeveloping the UW section of the trail (big pdf).