I was also worried about this before going to NYC in March of this year so I called the city to check it out. The woman on the phone asked how many people I would have in my crew. I replied "just the wife." This was followed by a short pause after which I was told I did not need a permit for a tripod. You only need a permit to shoot with a crew (other than "the wife") and other equipment.
I was advised to use common sense and not get in the way. Good advice because New Yorkers, while mostly very polite and helpful to tourists, are people who are generally on the go and they WILL trample you (mostly verbally) if you impede the flow, so to speak.
There is SO much to photograph in NYC. I had plans for some very specific photographs but wound up getting sidetracked continually and sometimes got no further than a few blocks in half a day. Some urban sociologists use the term "inexhaustible city" to describe places like Manhattan. No matter how much you see, there is always more.