Follow New Jersey laws as it relates to the transfer of a titled vehicle.

California DMV does not require in person notarized transfer according to the CA DMV website. You just need a signed vehicle Title and potentially a bill of sale if the seller is not the listed owner. Probably doesn't affect you.

That being said, it's unwise to purchase a private vehicle without having everything in front of you. Title, vehicle, owner. Your post is sort of hard to understand in terms of your exact situation.

Do a vehicle transfer in the state of your primary residence. And yes, once it's registered with plates/inspected then you can ride it anywhere in the US just like any other vehicle.