In San Francisco: The USS Hornet, The Golden Gate Bridge (of course), The Legion of Honor (if you are into really cool museums), Fishermans Wharf (tourist trap), Golden Gate Park (More of a walker than a hiker, but a nice place to visit. You can roller blade or bike ride and all sorts of other stuff) and Lombard Street (the crookedest street in the world). Driving in SF can be a real challenge so have a good map.
In Oakland: The Aviation Museum and Joaquin Miller Park (If you really want to go on a good hike with no crowds and beautiful scenery. The park has everything from pleasure walks to hikes that will make you cry. I live in Oakland and would be glad to take you if my schedule permits).
Farther North: Sequoia National Forest (biggest trees you'll ever see), Lake Tahoe and dozens of quaint small towns with their own charms, many from the Gold Rush era.
Farther South : Monterey and Carmel (The Monterey Bay Aquarium is fantastic, but over priced to get in. You saw part of in in STIV The Voyage Home. They used it for the whale's home), Yosemite (costs money to enter the park and is generally crowded in the summer, but the pass lasts seven days. Don't plan to camp unless you make prior arrangements as the waiting list is years long I think) and the Mammoth/Big Sur area (a good place to hike).