The closest airport is Florence, but for continental flights is Rome, another option is Pisa.

Cortona can be easily reached either by car or by train, being served by the Autostrada del Sole (Valdichiana exit), State Road SS 71 and the Milan-Florence-Rome railway.
 

There are two railway stations, Terontola-Cortona and Camucia-Cortona, on the track from Florence to Rome. Both are connected to Cortona by bus. The Terontola station is 11 km from Cortona and the Camucia station is 5 km from Cortona. The train schedules can be obtained from the official timetable of the Italian Railroad company, or alternatively at the DB, which also covers non Italian stations

Autostrada del Sole (State Highway A1, red on the map above):
"Valdichiana" Exit. Upon exiting the highway, take the Siena-Perugia State Road ( green on the map ) and continue towards Perugia. There are 3 possible turn-offs for Cortona: Cortona via Manzano, Cortona via Laurentana and Cortona via Strada Statale 71 (State Road 71). In this latter case, at Camucia there is a deviation for Cortona and Città di Castello (the Umbran-Cortonese Province Road).