Flight
Fly into SeaTac airport where you can either catch a shuttle to Whidbey OR rent a car and drive to Mukilteo where you'll catch the ferry
Flight
You can also fly into a smaller airport called Paine Field and make your way to the Mukilteo Ferry if you want a boat ride or you can drive all the way onto Whidbey Island.
Travel Note
Please refer to the schedule for shuttles from either airport in Seattle to Whidbey Island. Or you can take a bus if you fly into Paine Field: http://www.whidbeynewstimes.com/news/getting-to-paine-field-gets-easier/
House Or Rental
There are lots of places to stay on Whidbey Island. There are Inns and Bed and Breakfasts aplenty or if you wanted to rent a house, you can find them on Airbnb or VRBO.