The NYC permit allows you to own the gun in NYC. It only applies to NYC and probably NYS.
You then need to comply with the law in whatever state you plan to bring it to. NYS nor NYC can't put any restrictions on this as they do NOT have jurisdiction in those locations. For example, Florida could probably care less if you brought it down, but you'd need a Non-Resident Concealed carry license to carry.
If you want to shoot in NJ, you need to comply with NJ laws. NJ does NOT allow handguns from out of state except for NRA sanctioned events and to a range. (hunting doesn't apply as I don't know of any legal handgun hunting in this state) Depending on the officer involved and the DA you get stuck with you could be charged with a felony until it gets sorted out. Many of the cops here aren't familiar with the law and tend error on the side of arrest first and let the DA sort it out. As your destination is NJ, FOPA DOES NOT APPLY.
You can apply for an out of state NJ FID (firearms ID, aka purchaser card). Then you could legally buy handgun ammo and bring your firearm to NJ firing ranges and have less risk of a misunderstanding as long as you followed all the goober state rules
If you get an out of state FL concealed carry, you can carry your handgun in FL. ( Not NJ or or NY unless you like jail)