This guy always has great advice, I would use his route or the adventure cycling route:
http://www.crazyguyonabike.com/forum/board/message/?o=RrzKj&message_id=7160&v=DD&term=yosemite&context=allI say this having lived in Santa Cruz for 15 years and commuted by car to Yosemite about 100 times.
Leaving San Francisco, you can't ride across the bridge as shown on your mapmyride link - no bikes allowed on that huge busy highway bridge. You can leave San Francisco by bike to the north via the GOlden Gate Bridge (north Hwy 101) or by heading way south on the peninsula and riding across the Dumbarton Bridge (Hwy 84). Or you can ride down the coast to Santa Cruz.
You can also take BART (Bay Area Rapid Transit) which is a regional train system, from San Francisco (it does go to the airport) across the bay - this is very easy and cheap, and the bike can go on the train (except during rush hour). It costs just a few dollars to cross the bay. You don't need a reservation, you buy the ticket at the station. Here's the web site to learn more:
www.bart.govYou can also get a ferry boat ride across the bay, but I've never done that - I think the adventure cycling route recommends that, but I would just take BART, it's so easy, and the trains run very frequently.
Have fun!!