Hello Sarah and welcome to our community!
Trains are available to the various villages in the Cinque Terre. Times depend on which village you are going to. Here's an example of a timetable from Venice to Monterosso, from our website ItaliaRail:13
In this example, the travel times range from 5 1/2 hours to 6 hours, 12 minutes.
It's easy to search the train timetables at ItaliaRail13 to find the best time for your itinerary. Keep in mind that the timetables are released in approximately six-month groupings, so you won't be able to book more than six months in advance of your travel.
We have a page here in the community called Train Timetables14 where we post notifications when new timetables are released for booking. If you go there and click the blue "Follow" button, you'll get these notifications via email.
Hope this helps! Let us know if you have any other questions about planning your travel to Italy. Buon viaggio!