John F. Kennedy Jr., Carolyn Bessette, and her sister Lauren were flying to attend the wedding of his cousin Rory Kennedy in Hyannis Port, Massachusetts.

More specifically, their plan the night of July 16, 1999 was:

  1. Fly from Essex County Airport in New Jersey to Martha’s Vineyard, Massachusetts, to drop off Lauren Bessette, who had weekend plans there.
  1. Then continue on from Martha’s Vineyard to Hyannis Port so John and Carolyn could be there for Rory Kennedy’s wedding the next day, where John felt a strong family obligation to attend.

They never reached Martha’s Vineyard; the plane crashed into the Atlantic Ocean a few miles offshore during that first leg of the trip.

TL;DR: When people ask “where were JFK Jr. and Carolyn going,” they were en route to Martha’s Vineyard (first stop) and then on to Hyannis Port for Rory Kennedy’s wedding.

Information gathered from public forums or data available on the internet and portrayed here.