You can come out of Des Moines, south around Vashon and into Dockton. It is not as far in as Quartermaster, and has shore facilties. It is a state park, for boaters.
When you come out of Dockton, go up the west side of Vashon Island - Colvos Passage. There is less traffic and it has a very interesting feature - there is very little southerly current. On an ebb, you get the northerly push, but there is a shelf at the north end that limits the flood.
You can instead stay in scenic Gig Harbor, which is further past Dockton and Quartermaster, on the southern end of the Kitsap Peninsula. Stay at the city dock and walk around downtown. It is about 27 miles from GH to Bremerton City dock by boat.
If you want to get north and stay closer to Bremerton, there are places to anchor out in Sinclair Inlet, just watch for ferry and Navy traffic. Along the southern end of Sinclair, just east of the Annapolis ferry dock, there is an inlet below the veterans home (Retsil) that has good holding for anchoring out.
An alternative is to turn northwest out of Des Moines and go up to Elliott Bay. You can anchor out or go into Bell Harbor.
And one last option is to again go northwest out of Des Moines and head over to Eagle Harbor on Bainbridge.
Watch your timing going through Rich Passage. You can actually get 5-6 knots of current through there.
Joe