As far as I know, only one village can be reached from Genova on a regular basis, that is to say Monterosso. I suppose that the others may be connected to La Spezia through local trains, whereas Monterosso has a few InterCity trains (going from Genova to Pisa) which stop at its station.
The journey from Genova Brignole to Monterosso takes a little over one hour (be careful, in Genova there are two train stations, the other being Genova Piazza Principe, and they are not particularly close; if you need to go from one to the other, you had better catch a bus).
I checked the timetable for a midweek day in December and I came up with seven options of direct trains, leaving Genova as early as 7.54 in the morning or as late as 10.06 in the evening (look here for the timetable). The price is 9 euro for a direct one way ticket.
EDIT : Genova Brignole is the station from which all the trains going eastwards leave, whereas Genova Piazza Principe is the one which serves the western part of Liguria.
If you wish to get to another village in the Cinque Terre area, you may want to reach La Spezia and catch a local train from there. This however is more expensive and takes much longer.