By Plane: 

Cagliari is connected to major European cities via the Elmas International Airport, from which it is possible to reach the city center in several ways: every train arriving at Piazza Matteotti station in the city center makes a stop at the airport station; alternatively, taxi and car rental services are available.

By train:

Main Sardinian railway station, with suburban, local and regional  connections. All trains leaving the station make a stop at the Mario Mameli Airport in Cagliari - Elmas.

The train from the airport to the central station of Cagliari costs 1,3€ and the trip takes about 7 min. You can find more information and train schedules at IT - Trenitalia.