I would get into a taxi (there is a taxi stand outside each terminal) or use Supershuttle at JFK instead of using public transport. JFK is a long way out from midtown Manhattan and you would need to change trains at Jamaica station. IIRC Jamaica Station also has steps. With associated luggage and two children also in tow I would not use this for this particular part of your arrival into NY.
Super shuttle or Uber, although don’t know how Uber manage pickups from JFK. Also Lyft is like Uber but less morally bankrupt. Both are totally run of the mill options in the US. Trying to use public transport is a nightmare from JFK.
You are probably going to be coming out a lot later than you think, if you are going through the 'first time ESTA' channel. It takes a loooong time...you will be exhausted and just want to get there easily.
The taxis are very organised with marshalling and a fixed fare into Manhattan.
Honestly? I go to New York several times a year, and used to live there. I walk or get the metro every except the journey to and from JFK. It's $52.50 which i round up to $60 (for tip) and it is worth not having the stress of the train.
If you really insist..... it depends on which terminal you arrive at. If it's Terminal 7 (BA), then get the air train to Jamaica then change over to the Long Island Rail Road and head to towards Penn. Depending on where you're going on Broadway that you can walk over to 34th Penn and change to which ever line you need on the metro system.
I fairly recently used Super Shuttle to go from JFK to downtown, and it was really easy - you come out from customs, do the checkin on your phone, then wait in the Supershuttle area and they come and get you. Direct to the door and its $50 for all of you - possibly cheaper if you search for a discount code
Sorry, don't have any suggestions for this trip, but I do have a suggestion for the next time you travel to NYC - DO NOT FLY INTO JFK if you can possibly avoid it.
People have it in their heads that if they're going to Manhattan they have to fly into an airport that is in New York City, which JFK is (though it's in Queens, not Manhattan). Newark is right across the Hudson River, in New Jersey, but it isn't on people's radar. There are plenty of flights to and from the UK, they cost the same as flights in and out of JFK, and it's SO MUCH EASIER to get into Manhattan from Newark than JFK via the AirTrain.
Newark itself is a much easier airport to get around than JFK. I have only been once in JFK in the past probably 15 years - I'm a New Jersey native and still have family there - and it was an absolutely miserable experience. NEVER AGAIN!
I can tell you, if it helps, that the one time I went to JFK in the last 15 years I did get an Uber there from suburban New Jersey. Cost about $75, much, much, much less than a yellow cab would have cost.