Trains are much cheaper than booking flights. You can get an idea of the prices by looking at the prices for the next three months, but you can't book more than three months ahead.
We flew into Marco Polo and had pre-booked a water taxi transfer as a nice treat. Definitely an indulgence but I felt like Sienna Miller sitting in the back of a luxury white leather and wood, open topped water taxi. However you can get the water bus for a few euros. We stayed for four nights and then took a water taxi to the train station and caught the train to Rome. It was a long train journey, about five hours, but I think first class is worth it especially with your cases.
We stayed near the station in Rome and went sightseeing for three nights. The train to Naples and local train to Sorrento. We stayed in the Grand Hotel Capodimonte, up on the top floor in an amazing room wtih a stunning balcony. We splashed out on this trip because it was our honeymoon, we usually wouldn't get the best room in the hotel and take taxis everywhere.

We even hired a driver to take us along the Amalfi Coast which was a lovely day out. We also saw Capri, Mount Vesuvius, Pompeii, etc. We stayed six nights, but wished it was a couple longer, and hired a car to take us to Naples airport.