What mhendo said. Southwest flies out of Oakland, for instance. In fact, I would say that it’s slightly faster to get to the North Bay from Oakland than from SFO, which is actually south of the City in San Bruno. (Plus, San Francisco doesn’t have a through freeway system, so you have to drive on the city streets to get to the Golden Gate Bridge, which can be a nightmare, especially if you’re not familiar with the route.) I also understand there is an airport in Santa Rosa, but I have no idea where it is and I’ve never flown into or out of it. I can’t imagine it would be very cost efficient.
If you don’t want to rent a car, there’s a very good airport service that runs from Santa Rosa to both SFO and OAK. (Well, they have multiple busses, they don’t go to both airports on one trip.) Sonoma County Airporter.
(I grew up in Sonoma County. For some reason I’ve had cause to mention that twice on the SDMB today. It must be a very Sonoma-y day.)