There are 2 Tanger outlets. The one in Deer Park and the one in Riverhead.
For either one, take the Air Train from JFK to Jamaica Station.
For the Deer Park location, buy your train ticket first for Deer Park, and board an Eastbound train to Deer Park (which is on the Ronkonkoma Line). Once at that station, there is a shuttle service to "The Arches" which is what the location is referred to.
For the Riverhead location, buy your train ticket first for Riverhead. You'll have to take two trains. The first to Ronkonkoma, and the next to Riverhead. From there, catch a cab to Tanger Outlets.
Either way, it is going to be costly (figure at $25 - $35 per person) just for the rail transportation, let alone the time it will take to get there and back.
I live on Long Island.