Are you looking for a connection between the train station of Napoli Centrale and Sorrento?
There are few options to choose from. We can easily say that the cheapest way is the local train called Circumvesuviana.
The ride is pretty fast and it will take you about an hour to reach your destination. The cost is extremely reasonable to.
The trains can be "Accellerato", which will do all the stops along the way, or "Direttissimo" which will skip few stops, saving about 10 minutes on the total ride.
Lately a new option was added, the Campania Express, which uses the same tracks but have newer cars and skip more stops.
Make sure you double check which platform to use because the Circumvesuviana counts several different line of trains that connect hundreds of towns surrounding the Vesuvio's area. Jumping on the wrong train will cause you terrible delays and frustrations.
There are few things to consider before you decide to go ahead and use this option.
Most of all is the location of your final accommodation, if you are not staying nearby the train station in Sorrento and your place does not offer free shuttle, you will need a taxi once in Sorrento.
When you get a taxi you add an additional cost that can lead close to the cost of a full private car transfer from Naples to your final place. The second thing to consider when using the Circumvesuviana is the amount of the luggage you have, if you have large and heavy suitcases, the local train will not be easy, on board there is not a proper space to store it and you will end up standing for the entire trip.
Circumvesuviana is also a popular spot for pickpockets.
The most scenic option to reach Sorrento is probably the ferry, because this is the most complicated option, you will have to be prepared to switch different public transportation and you will need to have light luggage.
This is not the cheapest or shortest or most comfortable option, but is definitely an experience that will add few sights to your trip. First step from Napoli Centrale train station is the subway underneath Piazza Garibaldi, accessible directly from Napoli Centrale. This subway station has been recently opened and voted to be one of the most beautiful in Europe. Exit the subway at Municipio's stop and you will be steps away from the port. The port is close to the historical part of Naples and if you have spare time you could quickly check-out the Royal Castle, the San Carlo's Theater, Piazza del Plebiscito, Galleria Umberto, get a coffee at the most famous bar Gambrinus.
From the port of Naples, at the Beverello dock, the ferries connect to Capri, Ischia, Sorrento, etc.
The ride is beautiful and the view of Sorrento from the sea is one of my favorite.
Before you choose this solution you have to consider the location of your final accommodation and if they offer free shuttle.
It is not easy to navigate Sorrento if you do not know the area well. A place on the map can appear very close to the center but that could be, tiny road, several steps, limited access area, etc.
The other option which is the most comfortable is a private transfer by car. Is the best option if you have heavy luggage and do not to want to struggle switching vehicles, moving luggage back and forth. But most of all the driver will pick you up at the platform where your train arrives, help you with the luggage, drive you safely to Sorrento, no mater which is your final address you will be dropped off as close as possible to your accommodation. During your transfer to Sorrento the driver will share with you information about the area, he can suggest you how to have a tour of the Amalfi Coast, etc.
I hope this post helped you.