Unless something has changed drastically and very recently the NYPD no longer issues "target" permits. IIRC they now issue premis/target use permits. Again IIRC you may have a loaded/secured firearm in your home for self protection. If you want to go for target practice or some sporting event you may transport your handgun in a hard side locked container in a locked compartment of your car. (trunk) With any ammo also locked in a seperate container in a different compartment of your car. Funny thing is you can carry the gun in a locked container in your range bag with ammo in the same bag on the subway.
every pistol permit, and every purchase order comes with a detailed list of do's and donts. As far as NJ... once you cross the bridge or tunnel your subject to NY's laws.