You should have nice weather in Sept - end of summer so hopefully, sunny warm and low humidity. Look into NJ transit (Rail Road) from Penn Station to where you want to go on the NJ shore. NJ transit trains go to many of the shore towns (don't know where the "house" is!). You don't need a car to do anything within NYC (and by that, I mean within the 5 boroughs) - get a train map and plan your days accordingly. I'd aim for avoiding peak hour - will be much easier with children to avoid packed subway cars (and they are PACKED during peak hour). And NYC buses aren't bad either - a bit slower than the subway, but they do a good job of getting you from A to B.

Have a great time!