You would head north out of Vallejo before heading west on 37. 37 is a busy two lane with lots of heavy truck traffic. It has fairly broad shoulders and traverses the impressive San Pablo Bay Marsh. And I can say I have seen touring cyclists out here. While scenic, the traffic is not much fun (especially after traversing US 50 . . ). Once it returns to land at Sears Point, the road changes to a 4 lane highway. Shoulders continue to be fairly good, but as you reach US 101 you will be directed to side streets in Novato. The route from Novato to Sausalito is convoluted at best. Consider ordering the Marin County Bike Coalition map of the area to plan that. There are some dedicated bike trails on what remains of the former railroad right-of-way from Larkspur south.
You could also consider CA 12, a bit further north. Not much better though for the same reasons - scenic but heavy traffic. And there are stretches with zero shoulder forcing you to side roads which are great, but convoluted. And you're further north making it a longer run to rejoin the route to Sausalito.
Feel free to ask for any more detail on the area.
Santa Rosa, CA
Addendum: if you Google 'Vallejo to Sausalito' and select the bike route option, drag it over to 37 (it defaults to the ferry) it will take you to a logical bike route. From Sausalito you would continue on to the bridge and into the city.