Get a driver's license as quickly as possible. A driver's license is a universally accepted form of identification and life can be difficult without one.
Getting a license in California requires a written test followed by a driving test. The written test is pretty easy to fail so you will have to read and study the California driver's handbook then go to any DMV and take the written test. Allow 7 hours of study to become proficient enough to pass the test (3-4 nights of work).
Then you'll need to take the driving test: http://www.dmv.ca.gov/pubs/brochures/fast_facts/ffdl22.htm
California DMV does not differentiate between a manual (stick shift) license and an automatic license. So you can pass the test in your auto car and still drive a stick shift.
A California driver's license will allow you to drive anywhere in the US but driving law is a little bit different in each state.
Book all of your appointments at the DMV in advance and be sure to pick the first timeslot (e.g. 8am) otherwise you are in for a 1-3 hour wait. If you book early you can normally get in and out within half an hour. DMV is poorly regarded among Californian's as an example of a badly broken public service. They have very poor systems automation, terrible customer service and a terrible reputation. DMV will make you pine for the hi-tech nature of the NSW RTA.
Patrick Collins' experience:
Technically you need to have a US driver's license within 3 months of being in the US. However, several Aussies who never bothered to get their US driver's license at all and kept driving around as an international driver. I would not want to test this theory with a law enforcement official.
Bring an international license along
While I got by without one, I regret not getting an international license before I left Australia. A few insurance companies were happy to insure us with a CA license or an international one, but not with an Aussie license.