Angle Lake Cyclery is a blast from the past and worth the visit for nostalgia purposes alone. Its just a short bus ride from the airport. All the buses in Seattle-King County have bike racks.
I bought my current bicycle from Elliot Bay Bicycles, they are really grand folks, and are the home of Davidson Handbuilt Bicycles, which are objects of great cycling beauty. They also have lots of classic old racing bicycles that have been painstakingly restored. The Seattle airport express bus will let you off a couple of blocks away from Elliot Bay. They are a short down hill run to the ferry docks, but not to the Fauntleroy/Southworth dock, which will take you through the industrial/port area and heavy truck traffic. There is probably a Seattle bike map that would show you a much safer route. The ride from the Southworth dock up to Bremerton is not pleasant. I have done it several times.
If you are trying to get to the Olympic Peninsula (that is my recommendation) then take the Bremerton ferry from downtown. There is a brand new hotel and convention center right at the Bremerton ferry terminal and a fine Indian restaurant across the street, with dancing until 3:00 am (tell them Daniels friend sent you).
If you use Gregs Cycles then peddle north to the Edmonds/Kingston ferry dock and you will have your shortest shot to the Hood Canal Bridge and the Olympic Peninsula. Or keep riding north from Seattle to Mukilteo/Clinton-Keystone/Port Townsend ferries. That is definitely not the shortest way, but Port Townsend is a fun place to visit with a state park in town, lots of B&Bs and some restored historic hotels. It also has lots of great food and nightlife, and you might even see a Hollywood star or two having a hide-away vacation.
If on the other hand you are heading to Portland, Oregon and then out to the coast follow the STP route
http://www.cascade.org/EandR/stp/index.cfm and use Angle Lake Cyclery.
Western Flyer