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Go to the "Seattle Ferry Terminal" on Google maps (also called "Colman dock") via Uber, Lyft, Lightrail or car. WALK-ONS - Walk up stairs, and you can buy a ticket at a kiosk or by going up to the desk once you are up the stairs. There is an elevator as well, look for signs. Follow signs for the Bainbridge (not Bremerton) ferry. CARS - Drive into the ferry waiting area, you will pay for the car and the passengers to go to Bainbridge (not Bremerton) and they will direct you where to go.
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High-level: Take a ferry that leaves Bainbridge at least 3 to 3.5 hrs before your flight. Details: The ferry takes 35 minutes, it'll take about 15 min to leave the ferry and find a Lyft/Uber, and the drive from there is 20-30 minutes. Also, ferries can occasionally run behind. Thus, from the time the ferry leaves the dock, it will likely take 65 to 90 min to be at the airport.