To get your bike registered in CA you will have to spend $23 to get your registration and license plate. There are no renewal fees or stickers, it's good for the life of your bike. To register fill out sections 1,2 and 5 of form: Reg 230, sign it, send payment and mail it to the address on the form---->
https://www.dmv.ca.gov/portal/uploads/2020/03/reg230.pdf
When filling out the form use the serial number of your bicycle as the vehicle identification number (VIN). Also, since most of the 2 stroke 'china girl' engines have no serial number stamped on them when the DMV form asks for the engine serial number, it is acceptable to use the bicycle serial number and put an "E" in front of it in the spot on the form for engine number. "E" stands for engine. If the engine ever fails and you want to put a new one on your bike, you will not need to change your DMV registration. Leave the CA license plate spot blank, they will fill in a number once they assign one to you. Okay, that's it, send it in, wait 3 weeks, license plate and reg will arrive in the mail, no need to bring your bike to the DMV or anything, no inspections required.
In order to legally drive a motorized bicycle/moped on public roads in CA you will need to get a class 2 moped license or a class 1 motorcycle license, however, liability insurance is NOT required.