A 3rd party arranges busses to/from USMA to JFK, Laguardia, and Newark. In rare cases there are some to Newburgh. The schedules are published as the leave periods firm up. Previous years schedules are normally available, which can give you some idea of what to plan for.
These are timed near major events. You'll find that as a plebe, there are certain times/flights that make sense. You'll nearly always want a flight that arrives no later than mid-day on the day you are due back. DS normally shoots for 10-11AM arrival, which gives him a shot at two bus's.
When you can head out is harder to predict. Usually you end up with a 5 or 6pm flight the day leave starts, which allows for the last bus and transit/security time. Normally you can't leave until after last class, etc.
Parents clubs in the NE also arrange busses on major leave weekends, etc.
There are also private busses for Army/Navy if you want to take a spirit pass rather than ride down just for the game on the USMA bus.
Upperclassmen seem to shift away from the busses, as they have more access to cars and coordinate rides to the airport. For them, White plains (HPN) also becomes an option. To the SE, White Plains is usually cheaper than the NYC airports and is closer to USMA to boot. You can take a taxi to/from the airport, but it's not cheap.
As mentioned earlier, most cadets manage to find a way to fly even if they do not have parental support. Plan ahead, use the diary/buff card to see the candidates are for leave weekends. Work with your yuk on how to request a pass. Stay in good standing so you'll be able to take that pass!!! Do well and you'll get some performance passes!
Labor day weekend is the biggie for plebes. Columbus day is another, but there is a home game that weekend. So probably no pass until after the game.
If you cannot fly, there is always your sponsor's place, OCF trips, heading home with a buddie who lives in driving distance, etc.
Most importantly, get active in a club. There are dozens, and they are your best chance at getting off base and having fun away from leave periods. Some travel every 2-3 weeks!