In California you can apply for a driver's license without a SSN if you can prove legal presence (e.g. an F visa) but are ineligible for a SSN. Once you become eligible for a SSN you must obtain one and provide it to the DMV when you renew your license (which you will do frequently since California licenses for non-immigrants expire when their current legal presence does).
Once you have a license I don't believe a SSN is required to buy and register a car if you pay cash for it; you certainly won't get a loan without a SSN or ITIN since one of these is needed for a credit check. You'll need to make sure you can insure the car without a SSN, but I suspect this may not be a big problem since I don't recall having to provide mine to my own insurer.