I live in Upstate New York and might be able to give you some direction.
If you will be living in New York City, you will have an extremely tough if not impossible time getting a permit. Living upstate near the capital can take 4-6 months, as did mine. I would not recommend buying a handgun in Florida at this time if you plan on moving here in four months. You will not be allowed to bring it in the state until you have a New York state permit, as NYS does not recognize any other states' permit. Regardless if you want to hunt, use it for target practice or carry concealed, you will need a permit prior to purchasing a handgun or importing one that you already previously owned.
Hope that gives you some general guidance.
Here's a link to NYS's penal law 400 that states the specifics on handgun laws - http://ypdcrime.com/penal.law/article400.htm