last time I flew you were allowed 1 carryon each person holding a ticket and 2 suitcases per person holding a ticket. Holding a ticket meaning per ticket purchase just to clarify lol oh and for the lay over, you get off the plane and wait in the airport but you have to listen or ask where your connecting flight is. When i had a lay over they told us before we got off. Not sure if they still do that. You could bring your serger as a carry-on, but put it in a bag. You may have a problem though cuz don't sergers have a knife cutting thingy? That might be a security issue with all the crap going on these days. You can always call your airline and ask. If you can bring it then just pack one less carryon for 1 of your kids.
Things to bring with you...... well since your going to NY.... MONEY lol What part of Ny are you going to? I go to the city 3-4 times a year. It's ALOT of fun. Never been with kids so I can't tell you how that is but it is crowded in some areas but alot of people manage with strollers so its not a problem. Plus if you go to the city then definetly put those kids on one of those leash things or in a stroller. NY is ALOT of fun. not just the city lol if you go near the seaport bring noseplugs lol talk about NASTY!!!!
Hope I helped you some.
Don't worry you will be fine. just remember airplanes do shake a little bit at times so dont freak out. If anyone tells you scary airplane stories before you leave to go on your trip dont let them get to you. Landing is a piece of cake you cant even tell you land. At least i didnt when i went. I fly Delta and Midway. (not sure if Midway is still around) oh yeah and AA. Have a fun time on your trip. Oh wait one more thing..... if you are going to be near Westchester.... go to the Woodbury Outlets. Fun place, lots of good finds.