Naples to Cinque Terre

Ben, the best way is by train. You can explore your options here:

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

There are no non-stop (meaning with no changes) trains for this route so you will have at least two changes: one of them will likely be at Roma Termini and another in La Spezia. From there you'll take a regional train to your choice of 5 CT villages: Riomaggiore, Manarola, Corniglia, Vernazza or Monterosso.

You might also choose to take an Italo high-speed train from Naples to Rome if prices are lower for that leg than Trenitalia's high-speed options.

https://www.italotreno.it/en

Any way you shake it, it'll take a chunk of your day to get there: figure 6 hours minimum.

Here is a menu of choices so you can choose what is 'best'

https://www.rome2rio.com/map/Naples/Monterosso-al-Mare

Thank you very much for the tips. We'll take the train(s)!