what is the best way to navigate from the Cinque Terra to the Lamezia station or airport for a tour?
From which town in Cinque Terre?
Regional train to La Spezia then onward, probably change in Rome
Trenitalia
La Spezia to Lamezia
https://www.trenitalia.com/en.html
That's a very long distance away.
By train it is probably close to 10 hours, with multiple changes.
Given the distance I would consider flying from Pisa (or Genoa).
RyanAir flies to Lamezia (SUF) from either airport.