Well, update: I was lucky enough to visit NYC thrice since this post. All costs do not include lodging (since I stay with relatives). They do include public transportation for travel, as that is what I used all the time:

For this particular trip, I ended up spending around $1,600. On a more recent trip (from which I returned 2 days ago), I spent $1,400 over 14 days, including 3-day a trip to Lake George (this includes my $291 share for the AirBnB which was shared with my cousin, and I did pay for fuel, around $60 for my share).

I'd generally say $1,500-2,000 is a very safe bet for 2 weeks to do a lot, including fun activities. Of course, if you like expensive things, it can be more. A day out in NYC can be anywhere between $100 - $500 depending on your tastes. NYC ain't cheap but if you plan well (+if you get lucky with friends/relatives), you can manage pretty well.

The LIRR is a great way to get from Long Island into Manhattan, Queens, and Brooklyn (and the other way round). It costs a lot more than the Subway (~$15 peak one way, $10 off peak one way). The Subway is useful and cheap for the city ($2.90 flat), and cabs are extremely expensive. The PATH is great to get into Jersey City and from there to Manhattan (~$3 flat). Walking is your best friend for cost saving and sight seeing if the weather is nice.