Whats the Best Way to Fly from New York to Trieste?

You can fly Lufthansa from Newark into Munich and transfer to Lufthansa Regional for the flight to Trieste.
You cou;d also fly from JFK on Britsh Airways and connect for their flight into Venice. Then take public transportation (bus) the short distance to Trieste.

There is also an hourly train connection from Venice (Venezia Mestre station) to Trieste, taking some two hours.

Check flights to Venice (VCE) using www.kayak.com there will be many options, but I don't consider British the best since they only have one flight from Heathrow to Venice (most are from Gatwick).
Trieste has connections with major airlines only to Rome (Alitalia) or Munich (two flights with Lufthansa), all others are European low cost airlines. VENEZIA MESTRE station is near the Venice airport and within two hours you can be in the city center of Trieste.