I'm unsure of any tracks near NYC, at least not close by.
Look at the Del Val and NJ chapter BMW CCA.
http://www.delvalbmwcca.org/
http://www.njbmwcca.org/
They are very beginner-friendly, and have dedicated run groups for beginners. A good place to start is New Jersey Motorsports Park. The Lightning track there is spacious and easy to learn. NJ CCA is going there June 19-20:
https://www.motorsportreg.com/events...-jersey-431101
The price for that event is the going rate, not really going to get on track for any cheaper than that. Sometimes there are one-day events, but those are a waste, IMO, because you won't really get into it until the second day. Basically you'll want to go with a club like CCA, because they have rules, and it keeps the track safe and organized.
Since you are in NYC, you can drive down the night before and stay at a hotel (which I'd recommend), or just drive down real early the day of. You can get a hotel about 20 min away for not much money, I think the club has a discount at one of them.
You'll need an approved full-face helmet, and your car will need to be inspected. (See NJ CCA's site for the form, have it down at an indy shop or BMW dealer. Should be less than $100 to get the inspection done.) Other than that and a bunch of drinking water, you really don't need anything else to go.
Charles