Just a ‘heads up’ - ticks are rife in NJ and our friend’s daughter became very ill with Lyme Disease and I know of other people who caught it.
There isn’t really any open countryside as we know it in Europe; there are ‘Keep Out’ signs on private land and you have to go to township, county or State parks. It’s really not unusual to have bears roaming around the neighbourhood (we had a few in our town) but there are many in Morris County.
We used to get raccoons, opossums, turkeys and deer in our garden very frequently. Sadly at the South Mountain reservation on the edge of town it would be closed from time to time and the hunters would kill the deer with guns or crossbows :(
Many NJ towns have a “Rec” (recreation dept) which take kids from Middle School age + on Winter Saturday mornings to ski at Mt. Vernon, NJ or in the Poconos, PA.
Our family used to drive up to the Catskills in upstate NY to ski (but it’s not on the same level as the Swiss/French Alps and certainly no apres-ski: most people rent a house or cabin or just do day trips (bearing in mind the drinking age is 21).
In the Christmas holidays (called ‘Winter Break’ by the education dept) we would go with groups of friends and drive up to Lake Placid, NY Stowe or Killington, Vermont, or Mt. Tremblant, Quebec. There isn’t much in the way of powder snow though, it’s more icy in the north east ski areas.
School holidays are very long in the Summer, with one week off at Christmas and one week ‘Spring Break’ around Easter. Local public schools also have the odd days off in say, September for the Jewish New Year or other religious holidays where a significant number of students would otherwise be absent. Our town was like a ghost town in the Summer as most kids were away at Summer camps, or at the family vacation home on the Jersey Shore or in the mountains (with usually one parent commuting to work and joining them at the weekend).
Also FYI you have to pay your go on almost all NJ beaches! (Approx $7 + per day per person!) with some beache attendants checking your coolers for alcohol (banned - also at all parks too).