There are two international airpots nearby: Munich (MUC, 260 km) and Vienna (VIE, 300 km) with direct flights to all countries in Europe, as well as to most international destinations. Both airports are equally convenient to get to Schladming...
This is the most convenient option and takes about 4 hours, with only one change of trains in Vienna main station, and similar for Munich, or maybe two, depending on the time of travel. If booked well in advance, tickets are as cheap as EUR 19,- one way from Vienna to Schladming. The timetables for trains usually change in early December, try to book your tickets ASAP after that at https://tickets.oebb.at/en/ticket or by calling +43 (0)5 17 17!
...by a rented car:
Renting a car from the airport is a good possibility as well. If reserved in time, prices can be quite cheap. You can have a look at CarRentals, the journey will also take about 3-4 hours.
There are also two smaller airports in Salzburg and Graz, which are closer and even better connected to Schladming, but they don't serve many international destinations...
Have a nice travel !