If it is going to be a permanent change of residence
Well, good luck, and you have my sympathies. I left NY state nearly 35 years ago and haven't looked back. If you are moving to NYC for good, then if you want to keep your guns, expect to have to play by their rules, and pay for the privelege.
If you are only going to live there for a short time (even a few years) you might be better off finding someone trustworthy (and legally able) to store your guns with. That way there is a good chance you can have them if you get out of NYC.
As far as your gun rights, and the hoops you will have to jump through, if you must take them with you, you would be better off living outside of the boundaries of NYC, and commuting for work. New York has some very bad gun control laws compared to most of the rest of the country, and NYC is more restrictive than the general state laws. A New York State pistol permit is NOT valid inside NYC. Only a permit issued in NYC is valid there. And I don't know about long guns, but I imagine they have some special rules for them as well. I do know that anything they consider an assault weapon is forbidden completely. You need to check their laws very carefully, and do not rely on anyone but official documents for correct info.
If you are going to live there even for a few years, see if you can find an FFL dealer with a storage safe who would be willing to rent space for your guns. I doubt you would be able to legally use them in NYC anyway, so other than the cost of storage, what are you losing? If you have a few, consider the risk to them in NYC. If they are stolen, you are out. Even if recovered, I don't think you will ever get them back, the most you will get is a check from the city for what they think they are worth, and you likely won't even get that. Another worst case situation is that you go through all the legal BS they require, and pay the outrageous fees that demand, and then, sometime, something happens and the cops sieze them anyway. Even if you are found innocent, good luck ever seeing those guns again. And if you do manage to force their return through a court order (good luck on that one in NYC), they will be trashed. Most of the time even if a court orders their return, the cops won't do it. They will just cut you a check.
I don't mean to come off all doom and gloom, but NYC is one of the worst places in the nation for gunowners, because the Eloi will snub you if they find out, and the Morlocks will make you a priority target. NYC maintains the legal fiction that you can still own guns, as long as you comply with their rules. Good luck.
Someday, hopefully, you can move to a free country. 