The world (or at least the state) is your toastrack.
I only really know about the Bay Area firsthand, but here it is:-
Demo forest near Santa Cruz is good – if you were looking for a 1 day getaway and were in the SF Bay area it would probably be my choice.
Marin is appealing as it’s the start of everything – but its a lot of fire roads. There is a ‘trail center’ type thing at Taramancho near Fairfax, which is pretty good.
If you can get more time, or if your family like the mountains then Downieville is another top spot. There is bike hire and shuttles are available. It’s 2+ish hours from SF off Highway 80.
There is some good riding in the Napa/Sonoma areas but I’m not sure it would give you the ‘California’ experience you’re after. I know when I was in Moab, the Slickrock trail was top of my list (for my one day of freedom). There are arguably ‘better’ trails in Moab, but I wanted to ride the trails that I remember drooling over in MBUK in the mid 90’s.
If you are really tight for time and staying in SF, them Mt Diablo in the East Bay can offer some good (if occasionally cheeky) riding that is accessible form the BART (train thing). That’s my local ride, and I’d be happy to show you around. It’s fairly hot in the summer though, early start is best for both temperature and people.
The Mendocino area is supposed to be excellent for riding, but I’ve not been there yet, fixing that in October. I’m also hoping to get riding in Tahoe in September – so might have some more first hand info then.
Cheers