I just remembered that I need to ask about buying a vehicle from another person in a private sale.  If I recently bought a vehicle from a previous owner, is there anything I need to do in regards to this?
You need to get the car's taxes calculated. For that, you and the seller need both to fill a form DTF-802,  so the DMV can do the taxes maths.
Do you know about the DPPA. Is there a problem if the vehicle was registered for me in another state while I was living there?
There won't be any kind of problem, as long as it get the exempt from the california emission standard.