You should be okay with long guns, as they are not registered. If you have an FOID, you're legal- just make sure you have your FOID with you when you're transporting it.
With a handgun, don't risk it. NJ has a long and nasty history of being downright malicious towards gun folk- if you don't have the paperwork, don't transport the gun. It doesn't matter if you're of legal age, and it doesn't matter if you have a FOID card- if you do not have the records to prove you own a particular handgun, you could land in a world of hurt. Handguns are registered to one person and one person only, and if that person wants to let somebody else shoot their gun then
they need to transport it to and from the range.
The only exception in NJ handgun law is for guns purchased
before you were a resident of NJ; you are not required to register handguns already owned. However, no transfer of a handgun is legal
in NJ without a purchase permit. IIRC there is no provision whatsoever for lending or borrowing... an FFL holder can transport guns for a client or customer who is the actual owner, but that's it.
I'm not a lawyer and I don't play one on TV... nor did I stay at a Holiday Inn Express last night. Just another NJ gun guy who really wants to move to America.