There is a good train connection between Genoa and Savona. Depending on the type of train and the number of stops the ride takes 30 minutes to one hour. For a ticket you pay between € 4.50 and € 7.50 per person (one way, 2nd class). There is at least one train per hour. You can consult the train timetable in advance.
How do I get to the train station from the cruise port of Genoa?
If you are docked at the cruise port of Genoa and you want to make an excursion by train, then you best depart from the main railway station Stazione Principe. This is within walking distance of the cruise port. From Principe metro station, walk right. Follow the street and turn left. Soon you arrive at Piazza Acquaverde, where you can see the entrance of the station on your left.