Metro north goes to grand central (various methods walk, cab, bus, subway) to Penn station (New Jersey transit) goes directly to newark stop (make sure it’s not the city stop, it’s the airport stop (ewr)) then air train to any terminal. Amtrak goes from New Haven through penn station to ewr. You have to take air train at the end no matter what, New Jersey transit or Amtrak.

If you have an Amtrak or New Jersey transit ticket air train is free, use the same ticket. If you don’t then you have to buy an air train ticket (it’s been a while since I did this route and it might have changed?)

At any point once your in New York on metro north there are a bunch of different options for getting to Newark. Bus, taxi, shuttle, car service, New Jersey transit. Some factors in what you choose are time of day, cost and connivence, is it snowing or not, is it rush hour or a weekend, all good things to consider. Usually there is never an Amtrak that is anywhere close to the time I need it, and metro north to New Jersey transit is my go to. If it’s nice out and I don’t have a ton of luggage I walk between grand central and penn station, if it’s snowing I take the subway and if I have a lot of luggage I take a cab. The only time this hasn’t worked for me it was a blizzard and I ended up being late to the airport but they had already canceled my flight because of the storm, they shut down the rail lines shortly after that and I spent the night at the airport (weekend before Thanksgiving 2018), took almost 30 hours to get to San Diego. I think about that when I start to miss traveling…