Hi J., I am from Dallas but moved to New York City last August with my husband's job. We have a son who was 6 months old at the time we moved. Since my son was too young to really understand, I wanted to try to offer you the advice that you as an adult might benefit from most.
The East coast weather starts getting cold in November (50 degrees) and stays cold till mid-May (35-45 degrees on average) with last year having an occasionally (20 degree day). You will need to buy real winter coats (puffer coats) for you and the kids. You will need winter hats and mittens. Once you get all your winter wear on, the drop in temperature from 50 degrees to 35 degrees will feel normal.
I don't know where you will be living in New Jersey but I have heard that New Jersey is very green and beautiful. And... in the summer you will have amazing beaches (possibly less than 1 hour away from where you will be living).
If you are close to Manhattan, let me know I will give you all the kid friendly ideas I have like: Central Park Zoo, the Bronz Zoo, The Children's Museum in Manhattan (might be best for younger kids? They have a whole floor that is just Dora and Diego exhibits), The Natural History Museum (all kinds of animal exhibits and a whole floor just for dinosaur skeletons).
Also, if you travel by bus or subway, you should use the website hopstop.com. It's like mapquest except it gives you directions from point A to point B based on bus, subway, or driving directions.
We moved back to Dallas after being in NYC for exactly one year. I actually really enjoyed my time up there. I made tons of friends thru a playgroup started on a website called meetups.com. Give yourself 3-6 months to get adjusted to all the changes. Then, one day you will wake up and be amazed at all the opprotunities that you have up in New Jersey.
Also, Dutchwonderland might be fun to check out once you get settled.
Family Amusement Park
2249 Lincoln Highway East
Lancaster, PA 17602