You can either take a ferry over to Seattle and then drive north on I5 or you can drive north up to Port Townsend from Bremerton and take the ferry to Whidbey Island. The latter way would be considered the more scenic route with many state parks, bridges, and passes. Once you are on Whidbey Island you can drive over "Deception Pass" onto Fidalgo Island which is the home to Anacortes which is the home to the ferry landing that takes you into the San Juan Islands.
Here are the driving directions for the scenic route...
http://maps.google.com/maps?f=d&source=s_d&saddr=bremerton,+wa&daddr=friday+harbor,+wa&hl=en&geocode=&mra=ls&sll=37.0625,-95.677068&sspn=57.118084,114.257812&ie=UTF8&z=9Make sure to check out the ferry schedule before you start driving:
http://www.wsdot.wa.gov/ferries/schedules/current/Thanks for posting on the San Juan Islands message board./