You'd need to get from the airport to the Amtrak station (when I rode this I took the train to Seattle). Then you start your ride east by going WEST... I was compared to Wrong Way Corrigan by one person I met along the way.
Briefly, you bike and ferry north on the West sides of the Puget Sound, jumping up across several islands, approaching Anacortes from the south. It's about a 100 mile trip which I did in two days. Beware: there are hills, but that's a good warmup for WA-20 across the cascades.
Below are a few routes I got from the friendly folks on the Cascade Cycling Club forum back in 2005,
I blended them and had a great ride.
However you get to Anacortes, enjoy the trip!
Take Low level Spokane Street Bridge/Bike path
Left/North on Marginal Way which becomes Alaskan Way
Veer left into Myrtle Edwards park when Alaskan way turns right and becomes Broad and take the bike path and come out on 20th Ave West in the railroad yard near Fishermans Terminal.
Ride north on 20th and turn right/East on Dravus
Left onto 15th and right onto Nickerson (before the Ballard Bridge).
Head East on Nickerson about 1 mile and turn left to Cross the Fremont Bridge and then make a right on to 34th go about ½ mile and you will see the Burke Gilman Bike path starting on the sidewalk at the light at the intersection with Stone Way. Take the path, heading east and then north for about 12 miles to Kenmore – Tracy Owens Station/Logboom Park and make a left on to 61st Ave NE – head north, up the hill about 1 mile.
Left/West on 200th, which turn right and becomes 61st Ave NE and then in a few blocks as it crosses the county line it becomes 23rd Ave W.
Left/West on to 236th.
Right/North on Brier Road
Left/West on 214th
Right/North on 44th Ave W - you have an option of skipping the next few turns if you’re comfortable in heavy suburban/city traffic as the route rejoins 44th in about 2 miles.
Left/West on 212th
Right/North on 52nd Ave W
Right/East on 200th.
Left/North on 48th Ave W and go straight until the T
Right/East on 180th.
Left/North on 44th
Left/West on 168th
Right/North on 52nd
If you like scenic and lots of hills turn left on Picnic Point Road, if you don’t then continue as 52nd becomes Beverly Edmonds Park Road and then turn left on the Speedway (hwy 525) and follow it to the ferry dock.
If you picked the scenic route here are the directions:
Picnic Point Road – follow it as it veers right at the Y and down the hill, make a right on to Marine View Drive/Maplewood Ave go up the steep hill and turn left on to Marine View Drive and left when you re-intersect with Maplewood and follow it right as it curves at 116th and then left into the golf course development on to Saint Andrews Drive, up that big hill to Harbour Pointe Boulvard, and a left on Harbour Pt and then straight until the Speedway (hwy 525) and left down the hill to the ferry dock.
In Mukilteo get on the ferry to Clinton on Whidbey Island – it’s the only ferry J
From the Ferry dock, head up the hill about ½ mile to Bob Galbreath Road.
Right on Galbreath
Right on Wilkenson – Dead End with a bike path through
Left on Sandy Pt
Right on Camano
Left on 2nd which becomes Saratoga
Left on E Harbor, crosses 525 and becomes Fish Road
Right on Mutiny Bay Road
Left on Bush Point Road becomes Smugglers Cove Road
Possible Camping – South Whidbey State Park
Cross back over 525 making a right and an immediate left to get on to North Bluff Road
Right turn – North on 525
Right on Race Road
Right on Harrington Road
Right on Morris Road
Right on to Parker Road at the intersection with Hwy 20
Ride through Coupeville as Parker becomes 9th and then Madrona Way
Cross 20 veer left and then right to get on Libbey Rd
Right on Beach Rd which becomes Crosby
Left on Golf Course Rd
Right on Clover Valley which becomes Ault Field Rd
Left on Hwy 20 and cross Deception Pass Bridge – awesome whirlpools when the tidal currents are moving
Left on Rosario Rd
Left on Havekost Rd which becomes “A” Ave
At the T intersection with Oakes Ave the ferry dock to the San Juans will be to your left and downtown Anacortes to your right.
You can find ferry schedules at http://www.wsdot.wa.gov/ferries/. The Spring schedule ends in the middle of June so you may need to wait for the Summer schedule to be announced. But the two ferries you need run often enough that you won't be in dire straits if you miss one. The runs you need are Seattle-Bainbridge and Port Townsend-Keystone.
At the ferry landing on Bainbridge Island you roll directly onto hwy 305. This is a fine route across Bainbridge, with good shoulders most of the way. There are side roads but they are much more hilly and don't deliver much in the way of scenery, IMHO. Bikes are generally allowed off the ferry first, but after getting off any ferry, I recommend pulling over and letting all the cars blast off up the road. At Bainbridge you will find a nice little business district with food and coffee shops to the left at the first light.
305 takes you to Poulsbo, where I recommend you turn right on Forest Rock Lane and then immediately left on Little Valley Road. Little Valley changes to Big Valley Road where it crosses hwy 307. Big Valley is a nice quiet bucolic route. Then turn right on hwy 3.
Take hwy 3 to hwy 104, where you turn left and cross the Hood Canal Bridge. Narrow shoulders make this bridge the worst section of this route.
Once across the bridge you can choose either a scenic hilly route through Port Ludlow and Port Hadlock, or a less hilly less scenic (but still pretty) more direct route to Port Townsend.
For the most hills take the first right onto Paradise Bay Road. Bear right onto Oak Bay Road. Pass through Port Ludlow. In Port Hadlock turn right on Irondale Road. When Irondale ends turn right on hwy 19. Bear right onto hwy 20 and continue into Port Townsend. This route will likely have less traffic than the other alternative.
For less hills, continue straight off the Hood Canal Bridge. Yes there is a big hill, but trust me this is the easier way! Turn right on Beaver Valley Road/hwy 19. There is a visitor center here with water and toilets. Follow hwy 19, bear right onto hwy 20 and continue into Port Townsend.
Port Townsend is a cute town. It has restored Victorians, a thriving wooden boat scene, and a wonderful state park, Fort Worden, formerly a Coast Artillery site. I like The Public House for a meal and brew, and Elevated Ice Cream for that important food group.
On Whidbey Island, I think hwy 20 while not the most scenic choice is fine for cycling except through Oak Harbor. I was there at rush hour once and it was clear the drivers were in no mood to be nice to cyclists. Hopefully someone can route you around Oak Harbor.
At the north end of Whidbey there is a very high bridge across Deception Pass. Tidal currents here put on an amazing show. There is a state park on both sides of the Pass where you can spend some time down by the water.
I would recommend taking the Fauntleroy ferry to Southworth and riding to Port Orchard. At Port Orchard taking an old original turn of the century Mosquito fleet foot ferry to Bremerton. From B'ton ride north up the west side of the Puget Sound where it is more rural and scenic to Kingston. At Kingston take the ferry to Edmonds and at Edmonds continue north to Mukilteo. I can provide more detailed route information later. This was the first leg of a 10 day tour that my wife and I did a few years back that continued up Whidbey to Anacortes thru the San Jaun Islands to Victoria, Port Angeles, Port Townsend and returning to Seattle via the Bainbridge Island ferry. In Mukilteo we stayed at a great little B&B about 4 blocks from the ferry to Clinton. I can't remember the name of it now but can/will find it and forward it to your e-mail.
I also liike Ross' suggestion of riding up to Port Townsend and taking the Keystone ferry to Whidbey instead of crossing the sound at Kingston to Mukilteo you would continue north from Kingston to Pt.Townsend. This route would include the historic logging town Port Gamble (blackberry pie), of course Port Townsend (James House the first B&B in WA the el supremo!) a must see, and on Whidbey Island the first town in the state of Washington-Coupeville (great ice cream shop). Stop by the Captian Whidbey Inn, built in 1907 from Madronna wood logs, for a pot of Penn Cove mussels and a brew or stay the night on thier feather beds! Stop me now I'm having too much fun on this ride. Can I go too?