CA is big, about the same size as the UK. You'll feel at home in the northern coastal part(cool, foggy and wet).
Across the bay there is Muir Woods National monument, a small bit of the Redwoods, but it's close to SF.
Nothing like Redwood National park or the Avenue of the Giants up north. But you will still be in awe. I use to live not far and did cross country runs through the woods to the coast. Not a bad day.
Drive up to Mt Tam also from there. View is awesome. Then take the Fairfax Bolinas Rd to Fairfax from the top. Nice drive, great views.
See the Academy of Science, Eat out a lot too, great food.
If you go to Yosemite: Glacier point, take the drive over to Mono Lake using HYW 120. From there, you can go see the Bristlecone pine forest outside Bishop, about 2-3 hours from Mono lake. The back side of the sierras is really dramatic.
Also, see the Sequoia groves up in the mountains(Yosemite has a few, not as nice as Sequoia, but lots closer).
That's a week right there. Stop at local fruit stands as you drive around also. We got produce!
You can also drive down HYW 1 from SF to Monterey or up to the Sonoma Coast.
Either direction is awesome.
Check the tides and stop off and look at the macro algae in the tide pools.
Try Dungeness cracked crab if in season.
Also, the Monterey Bay Aquarium is an awesome place and location.
Make a day driving down there very early along the coast, hang out there.
Then drive a bit farther south(5-8 miles) to point Lobos State reserve.
You will not be disappointed:
Drive over MT Tam(I take this bike ride often)
Pt Lobos tide pools
Not a bad place to hang out.
Regards,
Tom Barr