The closest airport to Aiglon is Geneva, where there are frequent international flights to many parts of Europe and the rest of the world. Geneva is well served by rail, and the train to Aigle station, the correct stop for Aiglon, takes approximately 1 hour and 20 minutes. The Aigle train will be listed as going to "Sion and Brig". Do not worry that the information boards do not say that the train stops at Aigle. Visit Swiss railways website for timetables and connections.
From Aigle station, you may get a taxi to Aiglon, a journey of about 25 minutes, or a TPC bus, just outside the station. The stop for Aiglon is called Chesières-Aiglon College and is right next to the College. The bus journey takes about 30 minutes. You can find the bus timetable here (after clicking please choose the "Aigle - Villars-sur-Ollon" link).
Swiss railways use an integrated ticketing system and it is possible to buy a ticket from Geneva Airport to Chesières-Villars, which is valid for both bus and train.
If you are coming from the north, via Lausanne, follow directions along the A9 autoroute in the direction of Simplon and Grand-Saint-Bernard.
If you are coming from the south, via Martigny, follow directions along the A9 autoroute in the direction of Lausanne.
Take the Aigle/Leysin/Ollon exit, and follow signs to Ollon and Chesières-Villars (this involves bearing right at the traffic lights 300 metres after exiting the motorway, and turning left after the BP garage about 2 kilometres later). The road continues for 10 kilometres uphill past Les Combes and through Huémoz.
On arrival in Chesières-Villars, continue for two hairpin bends and you will see the large blue Aiglon signs after about 450 metres on the left-hand side.