I've done NY-Boston and NY-Philadelphia a bunch. It is really easy. Getting to the station 30 mins early is all you need to do. Pretty straightforward -- print your ticket out at home or at a kiosk at the station (no need to wait in the long line to talk to an agent at the window, unless you have a problem), and watch the big board for the track announcement. There is no security to go through or anything like that.
A few other notes:
(1) It will probably be cheaper for you all to fly to NC, and it will certainly be faster.
(2) The portion of the trip along the CT coastline is really pretty, always a highlight for me.
(3) The food is pretty meh, and overpriced, though my dad happens to love the cheeseburgers. You can bring your own food onto the train.
(4) Not sure if this is of interest to you or not, but Amtrak does serve alcohol (again, over priced). I believe you can bring your own with you (at least that seems to be what everyone in NY does), but you might want to confirm.
(5) There is a "quiet car" all the way in the front right behind the business class car, and given you are with a 2-year old you probably won't be welcome there -- just letting you know ahead of time
Best of luck!!