Cinque Terra to Rome

Train.
First you have to get to La Spezia and the time that will take depends on where you're starting from in the CT.
From La Spezia to Rome, I think the travel time might vary between 3 to 4.5 hours, depending on how much you want to pay, the type of train you take (and want to pay for), etc.

Check this website for more info:
http://www.rome2rio.com/

Also check the official Italian train website, or for schedules only the easier to use bahn German train website.

Dk,

You can also check the Italian train website: http://www.trenitalia.com

There are so many different train options to get from CT to Rome but the fastest is the one where you take a Regionale Train to La Spezia then connect to a freccia ( high-speed) train and this would take around 3Hrs & 50 mins.

Check this for an example (leaving Monterosso @12:42 arriving @ Roma Termini @ 16:32): https://www.lefrecce.it/B2CWeb/search.do?parameter=searchOutputViewer&cFID=0OQNO1sZbP2n

dk,

Travel from the Cinque Terre to Rome is actually very easy. As the others have mentioned, the first step will be to get from whichever of the five towns you're staying in to La Spezia Centrale. There are numerous trains from there to Roma Termini each day, with the quickest ones being the direct (no changes) Freccia or InterCity trains which have a travel time of about 3H:45M. Some of the direct departures are at 08:17, 10:06, 13:14 and 13:56, and there are other trains with changes on that route.

Note that the "faster" trains such as Freccia or InterCity have compulsory seat reservations, which will be provided with the tickets. These tickets are specific to train, date and departure time, so can only be used on the one train listed on the ticket. Those without valid reservations for the train they're riding on will face hefty fines, which will be collected on the spot.