As already mentioned, you are moving to a place that has one of the highest costs of living in the entire U.S.
If you're worried about the price of moving a few possessions, you probably can't afford the Bay Area. On one hand, I would suggest rethinking your plans; on the other hand, there's no better time to follow your dreams than when you're young, single, and already broke
Good luck!
Oh - and to answer your question - I would rent a U-Haul trailer and tow it to Cali with your car.