I would like to visit my sister who lives there, and I'd love to climb with her!
Mt Diablo (1 hr drive east)
Mount Saint Helena (1 hr drive north)
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If you're not afraid of driving a couple km, there is a fair amount of routes in Castle Rock State Park near San Jose, and the landscape is quite beautiful!
Go check out Mission Cliffs for the best indoor climbing and climbing community in SF. They can definitely direct you to outdoor spots and others that "know the ropes" in the city. ;)