Travel from Siena to Sorrento

Definitely take the train. You'll actually take the train to Naples (Napoli) and then transfer to the regional train to get to Sorrento. Have a great trip!

There is no such thing as cheapest and fastest on public transportation in Italy. The faster you can go on the ground or water, the more expensive it is. You do have choices. From Siena to Rome, you can take the SENA bus. Or, you can take the train from Siena to Naples. It will be a cheap and slower Regionale train or bus out to Chiusi where you can change to another slower train to Rome where you can change to a more expensive higher-speed train to Naples. There are no runs without changes of some type from Siena to Naples.

Once in Naples, its back to the very cheap and slower train from Naples to Sorrento. Its the only one so you have no choice. Or, for more money, you can go the ferry docks about a mile away and take the hydrofoil to Sorrento. This is faster but again, more expensive. You are looking at about 7.0hrs travel time.

With a bit more specificity on your travel plans, we can provide you with better options and fares.