Without a car Bainbridge is going to be a lot easier. It also gives you easy access to the ferry system. From Seattle the Kitsap fast ferries can get you to Kingston (highly recommend Sourdough Willy's Pizzeria) or Bremerton if you're at all interested in seeing them. Or there are the Argosy cruises.

You might like Fort Ward or the Bloedel Reserve on Bainbridge. Kitsap Transit can get you to North Kitsap Heritage Park if you wanted to do some hiking there.

If you do get out here with a car some time Orcas Island is the best.