I was a full-time student from January-May 2016, then I started my job in Florida in September 2016. I want to make sure I'm filing to the right state
If you had any income at all during the time that you lived in New Jersey you will have to file a part-year resident return for New Jersey. But you will only pay New Jersey tax on the income you earned while you lived there, not on the income from your job in Florida.
Florida has no individual income tax, so you do not have to file a state tax return for Florida.