Sorrento to Cinque Terre transportation

Take the local train from Sorrento to Naples. From Naples there should be a direct train to La Spezia or possibly a couple of the Cinque Terre towns. This will take most of a day, or maybe 6 or 7 hours. The train ride is not as scenic as you might imagine since you see very little of the sea.

We take the train from Roma to La Spezia annually. It is a nice trip with SOME sea views. I do believe you will take a Frecciarossa from Napoli to Roma then change to a Frecciabianca. You should be able to plan it out on the Trenitalia site, knowing the Sorrento to Napoli leg is the local CIrcumvesuviana. As Swan says, it will take much of the day. I'd say, without looking it up, 7 hours on trains plus waiting/changing time. A car will take you MUCH longer however, as the trains are very fast versus a car. And the train is relaxing. You can read, sleep, eat. Bring food onto the Freccibianca, though. There's not a lot on offer on board.

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Are there any places other than Sorrento and the Cinque Terre that you are planning on visiting?

This looks like your best connections from Naples (after the Circumvesuviana from Sorrento) ...
1) FR 9532 Dep 12:00 NAPOLI CENTRALE (Italy) Arr 13:05 ROMA TERMINI (Italy)
2) FB 9774 Dep 13:57 ROMA TERMINI (Italy) Arr 17:13 LA SPEZIA CENTRALE (Italy)
3) RE 11390 Dep 17:28 LA SPEZIA CENTRALE (Italy) Arr 17:51 MONTEROSSO (Italy)

Definitely take a train. Relax see the country. There are various options for the rail. Check out Rick Steves. recommendations