Upvote:1
With 2 adults and 2 kids, I agree with @Karlson that renting a car would be the cheapest option. A road trip between NYC and Greensboro includes great destinations like Philadelphia, Baltimore and Washington DC - you can even think of breaking your journey into 2-3 days. These cities have plenty options to explore for children and adults alike. Driving, even at for extended stretches is pleasant, given the highway quality and resources along the way.
If you do not plan to go back and forth between your destinations, you should explore renting from a national chain such as Hertz or Enterprise, which specialize in one-way rentals.
Alternately, if you look early enough, you may get cheap air tickets as well (lower than train fares).
Upvote:3
The cheapest way would most certainly be renting a car. It stands to reason that you would need a car in Greensboro so you might as well have one of your own so you are not dependent on your relatives to provide you with one and the cost of renting one in Greensboro would be about the same as renting it in NYC if you are returning it to the same rental place.
Having said all of that I would consider one very crucial factor in this: Are your kids given that you live in Israel prepared to travel 10 hours non-stop or if you count for stops 12-13 hours from NYC to Greensboro in the car? Most people who came to visit me(from Israel that is) consider this like flying to the moon. If not you would need to fly to Raleigh-Durham and rent a car and travel from there.
Amtrak might be cheaper then a plane but it's 12 hours based on the schedule and you will still need a car in Greensboro.