You need a minimum of a Class "C" license, wear a helmet, and have insurance, although the law doesn't explain that too clearly.
You can expect everyone from DMV to the cops not to know anything about mopeds. As long as you ride responsibly nobody hassles you.
Best kept California moped secret:
If you need to register a moped that has not been registered before, go to DMV and ask for form REG.230, Motorized Bicycle. Take it home, fill it out and send them $6.00. In a couple of weeks you will get a license plate and registration good for as long as you own it!
Don't let the Do-Do's try to register it as a motor-driven cycle, or you'll have to pay for tags every year. Just get that form and get out of there!
That's how I did it recently with the two Solexes.