While staying in San Francisco proper you will be better off without a car, as parking fees are high.

Yes, you can drive from San Francisco to Yosemite. It is approximately 200 miles and will take you about 4 hours, as long as you don't leave the city on a Friday after noon when everyone else is escaping for the weekend. But, it is not worth the drive unless you stay in Yosemite for two nights, IMHO. Lodging at this late date will be hard to secure, but you can try at yosemitepark.com.

Top attractions in SF and LA? There are many great suggestions on this forum if you do a search. What are your interests? Where are you from? Both cities have much to offer you.

Vania,
Yosemite by car, as Susan said, is about a 4 hour drive, could be 5 depending on the traffic. If you plan on Yosemite, I would call ASAP for reservations (www.yosemitepark.com). Other side trips I would include would be Monterey and Carmel (www.monterey.com) You could spend at least 2-3 days there. The Central Coast of California from Monterey going south offers some of the best coastal scenery in the state. I highly recommend the Monterey Bay Aquarium, 17 Mile Drive, Cannery Row, and Fisherman's Wharf. If you go a little further south along Highway 1, I would stop in San Simeon to see Hearst Castle (www.hearst-castle.org) Magnificent estate and great scenery as well. It's perched on top of a hill overlooking the ocean. San Jose and Silicon Valley would also be a worthwhile side trip. I go there on business on a regular basis.
(www.sanjose.org) San Jose is mostly known as a high tech mecca, however the city as its suburbs have plenty to offer (Restaurants, Museums, Theater, Ballet, Opera, etc.) I would not miss the Tech Museum of Innovation in Downtown SJ. While in San Jose a side trip to the suburb of Los Gatos , specifically Old Town, has a very quaint atmosphere. North of SJ I would recommend a tour of the Filoli Estate (www.filoli.org) For those that used to watch the show Dynasty back in the 1980's, this is the place where all of the outside scenes were filmed. I have been there once and it is a great place to see.