By air
Turin Airport
Turin Airport is the closest to the city of Turin. If you land in Turin, you can then take either a bus (45 minutes trip into the city, A/R ticket is 14 €) or a taxi to the city centre. As of Monday, 22th January there is also a train connecting the airport to the stations of Torino Porta Susa and Torino Lingotto managed by Trenitalia (31 minutes trip into Porta Susa, single ticket is 3.60 €).
Bergamo Airport
If you cannot land in Turin, Bergamo Airport is the destination of many low-cost airlines. From there, you can take a bus with Flixbus or with Itabus that will take you to Turin.
Malpensa Airport
The airport of Malpensa in Milan is on of the main national hubs, and most airlines will fly there. If you need to land in Malpensa, you can take a bus with Flixbus or with Arriva that will take you to Turin. Alternatively, you can take a bus or a train to Milano Centrale train station and then a train (fast or slow) from there to Torino Porta Nuova train station.
By train
If you are coming from within Italy, both Trenitalia and Italo have fast train services that stop at the two main train stations in Turin.
If you are coming from Spain, check the Trainline website for possible connections to Turin.
By bus
Flixbus offers service from many Italian and Spanish cities toward Turin.