Kitzbühel

The closest airports are in Salzburg, Munich, Vienna - in this order. And yes, there are trains to make the journey. You are in Europe. I would choose the airport based on price to fly there and also the one which has direct flights from New York. My educated guess is that will be Munich and Vienna, I don't know about Salzburg.

We flew in to Salzburg then took OBB regional train to Fiberbrunn, a town in the Kitzbühel District. Te benefit of that route is being able to spend a few days in Salzburg, not a bad place to adjust to time zone change.
http://www.oebb.at/en/

PS For 2 to 5 persons, the OBB Einfach Raus tickets are a good deal. No advanced purchase required. We bought tickets at the staffed ticket counter in Salzburg a hour before departure.
http://www.oebb.at/en/angebote-ermaessigungen/einfach-raus-ticket

The easiest way is to fly non-stop into London Gatwick, and then catch EasyJet.com into Innsbruck.
Innsbruck is just a short distance west of Kitzbühel.

When going to such an incredibly beautiful place, the best way to see the scenery and Tirol is with a rental car. I once flew into Innsbruck for a ski week and it was not nearly as enjoyable as the many trips I've made there with a car.

Hi,

Between the airports of Salzburg and Vienna, there is also Linz airport.

But coming from New York, I'd take a nonstop to Munich. Train from there is pretty painless. Change trains at Munich Hauptbahnhof (or Ostbahnhof) and Woergel.

I appreciate all the advice. As always, you guys are great!