Are you 25+? If yes, rental cars are moderately expensive (roughly $175/week). If no, they're ridiculously expensive (roughly $350/week). You CAN find them cheaper, but that's a solid average. I'm not even sure how car rental works if you have international ID, so I hope you've maintained a US state ID. The thing that will kill you is probably gas, unless you drive in Japan in which case you might just be relieved that gas is so cheap. It's running about $3/gal in CA right now. Word to the wise about car rental: make sure you KNOW of some place to park. Parking is kind of ridiculously bad in most places, or else it costs a premium. Sometimes both.
If you're staying in/around the Bay Area, the price of public transit will depend on how long you're staying and where you want to go. Bus trips on AC Transit (much of the East Bay) run $2 one-way or $80 for an unlimited 30-day pass. The MUNI (San Francisco) is also $2 per one-way trip, but if you're planning to use it a lot or be here a while you can buy an unlimited pass for $60 or $70 (the more expensive one includes in-city BART transit). The BART, which is the best way to travel long distances, starts at $1.75 one-way and goes up to around $6 one-way. EDITED TO ADD: SFO-to-anywhere or anywhere-to-SFO tickets on BART are more expensive than other fares because the BART train goes almost directly to the terminals. I believe it's $7+ one-way right now.
I can give more detailed information if you know more about how long and where you are going to stay and what kind of places you might be looking to go.
This goes for anyone, by the way! I've lived in the East Bay for a few years now, and my mother's family lived in North Bay for 30+ years, so I'm reasonably familiar with the area.