Transportation Cinque Terra to Carrara?

Train will get you there. Cinque Terre is not a station. Use the name of the station you will use.

You can do it by train:

http://www.trenitalia.com/tcom-en

You'll need to enter your CT town of departure, choose "Carrara-Avenza" from the "to" dropdown and pick an hour of the day. The time it'll take you to get there depends on departure city, hour of departure, and whether it'll involve a change of train in La Spezia or another station.

For instance, choosing a random day departing from Monterosso, I'm seeing one direct train at 8:45 that takes 45 minutes. Later trains take longer and involve a change. That direct train is not an option from Vernazza: you'll change trains once.

These are inexpensive tickets you don't need to purchase in advance of your trip: do it at the station the morning of your journey, or stop by the station and buy them the day before, if that's more reassuring.

The train station at Monterosso is sometimes very crowded. Buy train tix a day in advance if possible.