best way to get from rome to cinque de terre?

Cheapest day of travel would be 43.70 euro with an IC train to La Spezia followed by a Regionale to the CT. Takes 4 hours 20 minutes or so. With advance purchase nonrefundable tickets, you could get it below 30 euro.

And just FYI, it's the Cinque Terre. It translates as the "Five Lands." No "de".

There is a high speed train that connects Roma Termini station to Monterosso with 1 train change in La Spezia. If you know the date you travel and time and book in advance you can get some cheap fares of 9, 19, 24 euro for the trip. I used Oct 2nd as an example on Trenitalia.com and fares ranged from 11.70 to 27.90. The Freccia trains depart every 2 hours.

They also have other options for 44 euro that gets you into town 20 minutes less then the option above but you have 3 train changes. High speed train from Rome to Florence, change train a train to La Spezia, then change train to the village.