*edit- Disclaimer: I am not lawyer. I just happen to be an NJ gun owner who's looked into this particular issue on account of storing guns at homes in both states.
If you are transporting a firearm from your residence in NJ to an out-of-state location where it is legal to possess that firearm, then you are covered by federal law. It doesn't matter if you're taking it to another residence or a range in PA... or just driving across the border to sit parked in your car with the gun locked in your trunk while making faces across the river before driving it back home. PA doesn't have the transport restrictions that NJ has, and it isn't any of NJ's business what you do with your property while not in NJ. Federal law protects the transportation of a firearm from one place where it is legal to possess (your house in NJ) to another (the whole state of PA).
Transport within NJ is sticky because it is illegal to possess a gun without a valid NJ carry permit anywhere except your home, your gunsmith's place of business, and the range. There is an exception for transport between those places, but if you make any stops along the way you're cooked... that is not the case on the other side of the river. In PA, it is legal to transport a firearm anywhere except the usual list of places where firearms are prohibited- schools, government buildings, military posts, etc. So long as it stays locked in the trunk of your car, you can stop and grab lunch on the way to the range, or swing by your friend's house... all the things you can't do in NJ.
PA also does not have registration or a purchaser ID card like NJ does; since you already own the gun you intend to store at your place in PA, nothing else needs to be done. It's as simple as driving it across the river.
Also, both resident and non-resident FL carry permits are valid in PA.... not to mention ~30 other states. It's a good investment to make- just be mindful that a FL carry permit (or any other out-of-state permit) will NOT be valid in NJ.