If you arrive at Genova airport ("Genoa" in English), you need to get a bus called AMT Genova that will take you to the train station of Genova Principe. You can buy tickets on board (the cost is 6 euros).
From there, there are direct trains to Alessandria whose destination will be listed as Torino Porta Nuova. There are trains about every hour and this trip takes about one hour.
