You will get the drift that any off roading around the Bay Area is hugely limited, thanks to legislation keeping motorized vehicles off most of it...
Doc Wong does a good clinic regularly down at Clear Creek, but its challenging on a larger dual sport that said definetely doable. Expect to drop your bike though as some of the routes are knarly. The one day courses are free (Google Doc Wong riding). Picture of a class below (you dont get many GS's going!
EDIT - f**k Smugmug!
The best thing to do is head off over the Sierra's into Nevada. Lots of off roading there, everything from easy trails to single track. The eastern Sierra's also have some great trails to explore.