Train travel to Cinque Terre?

yes, from the north, south and east. go to trenitalia.com for schedules and fare information from your point of departure.

The national rail lines of Italy connects to Cinque Terre
Cinque Terre has 5 towns each have train stations.

Where are you traveling to in Cinque Terre?

You’ll need to type the proper town name ( Riomaggiore, Manarola, Corniglia, Vernazza, and Monterosso)
when searching train schedules on Trenitalia.com

Enjoy your trip!

Cinque Terre is, as the name suggests, an area containing 5 towns: Monterosso al Mare, Vernazza, Corniglia, Manarola, and Riomaggiore.
They each have a station. Rail is not only the best way to get to the 5 towns, it is also the best way to travel between them. They have minimal road access,. The other options, apart from trains, is walking between towns, or a ferry.

You can look up train times and get prices on the Trenitalia website: https://www.trenitalia.com/en.html
Where are you starting from to get to the Cinque Terre?

Oh dear, I was in such a hurry I was not clear. I know there is train service within the Cinque Terre. Just trying to figure out how to get to Cinque Terre. Thank you [to the poster who told me about TrenItalia.

I know there is train service within the Cinque Terre. Just trying to figure out how to get to Cinque Terre.

Yes, it is not disconnected from the rest of the Italian railway network. The trains come from somewhere before calling at the 5 towns (La Spezia), and continue afterwards (Genova) - or vice versa if southbound.
Depending whether you are coming from the north or the south, you would go through one of those two places.
If you put in your starting point, and your choice of the 5 towns, into the Trenitalia website, it will give you all the connections.

As the others have mentioned, there are frequent rail connections to the Cinque Terre. In order to provide more detailed information, it would help to know where you'll be travelling from prior to the CT, and where you'll be heading after that. In any case, your best resource for information is the Trenitalia website. You can also use Bahn.de for planning, but can't buy tickets on that site.

When is this trip taking place?