PA's laws are a little friendlier.
For specific information on New Jersey, visit
www.evannappen.com. He's an attorney who specializes in NJ firearms law.
The way I currently understand it, a resident must carry the NJ-issued Firearms ID card to transport firearms to and from the range. This proves that you are legally allowed to be in possession.
While I do not know if it's a requirement, it's a good idea to bring your copy of your handgun purchase permit.
As velocette says, however, the laws are what the local badge head says they are. If the local police decide to make up laws as they go along, there is NO accountability on their part.