At a minimum, you should have a helmet, patch kit, spare tube, tire levers, and mini pump or CO2 canisters. And a floor pump for pumping your tires up before you leave the house.

If you plan to leave the bike anyplace for any length of time in NYC, you need a good lock. Something like a Kryptonite New York lock, or an Abus. I mostly ride in rural and suburban Wisconsin so I use a cheaper lock on most of my rides, but if I leave my bike in Milwaukee, I use a Kryptonite lock.

Then the rest is up to you. Shorts, tights, jersey, base layers, windproof or rain jacket, thermal jacket, gloves, clipless pedals and shoes? Some kind of eye protection? Sure. Lights for riding at night? fenders for riding in the rain? If recreational cycling becomes your thing, you will probably acquire all this stuff and more in time.