Yes, everyone driving on public roads in CA must have a valid learner's permit or driver's license with them. CA acknowledges foreign licenses, but not foreign or out-of-state permits.
If you don't yet have a permit, check out these resources:
If you are under 17.5 years of age, you will first need to complete an Online Driver's Education course like the one on our website.
If you have completed that course, or are over 17.5 years of age. Here's a helpful link to prepare for the knowledge test.