from every law I have read, if the gun is legally owned in your current state and conforms to all laws in the state you move to you can legally bring it. If your state requires mandatory registration then you have to register it, if not don't worry about it. No rifle or shotgun legal in NJ requires registration in NJ so no worries there, the only thing i'm not 100% on is if you have to register the handgun, I don't think so but I would call the NJSP and ask for sure. Say you currently reside in PA and will be moving to NJ in the future and are curious if you have to register your handgun when you move here. They should be able to give you a better answer than me.
This guy is the number 1 recommended firearms law layer in NJ on all the boards i'm a member of, when in doubt, call him and ask.
http://www.jackventurilaw.com/Practi...ms-Weapons.asp