The Redwood National Forest is near/in Crescent City, CA. Here's a ton of information, I don't think its visited often since its sooo remote (I think i've only been there once when very young) If you're coming from LA its like 6 hours NORTH of SF:
http://www.nps.gov/redw/
Gypsy: Sequoia National Forest, which has one type of Redwood, the Sequoia (largest trees in the world!), is in the middle of nowhere (eek people from Visalia are going to kill me) by the Sierra/Nevadas. I'd say its about 5 hours from SF or 4 hours from LA. Pros: Much much less crowded than Yosemite (still crowded though). Also, has the yummyest ice cream, and has river rafting. Has these huge jaw droppin trees. Cons: It gets HOT, I mean like 90 degrees hot during the summer, and is less tourist friendly (roads are more windy, less cabins & campgrounds etc.).
More info here:
http://www.nps.gov/seki/
As to another city....unless you can get a flight into Sacramento, to access Yosemite. I would just say stick with your original plan. There's tons of outdoorsy things to do in the Bay Area, honest, and the forest by Carmel or Muir woods is really really pretty and has the same huge tree effect.
Oh another place to think about, before I get is Lake Tahoe which is supremely beautiful. I think you may be able to fly in there?
Hope that helps!!! I've lived in CA all my life and love travelling/exploring. I also was in Sequoia last summer, whereas Yosemite has been..gosh...6, 7 years.