Trains in Italy cost different amounts depending on the length of the journey, so if you're in a hurry it's a bit more expensive but there are plenty of trains going between the places you want to visit. Have a look on trenitalia.it which can give you times as well as prices.

If you're looking to save either time or money you could always get a night train between Rome and Naples or Naples and Florence.

You can definitely get a train easily from Rome to Naples. But whatever you do, if you take the night trains, make sure that you get a reservation. They are absolutely jam-packed, as in you could be sitting on the floors or standing. It may not be worth the cheaper cost of the night train. I'd recommend the Eurostar from Naples to Florence. It may be a bit more expensive, but if your experience is anything like mine was, this will be much more comfortable and way less stressful. You can also get there directly instead of having to change trains anywhere. It will take less time, so you'll have more opportunities to sight see. The trains leave throughout the day and take between 3 and 4 hours.

Hi Gulutz:

Driving in Rome is madness – and to enter or even park in Florence is not always easy either. Nevertheless I would recommend renting a car to do the drive between Rome and Florence, which is really quite nice. Florence stays at the very “entrance-steps” of the Tuscany Apennines Mountains, and when driving you will see many little charming villages, especially if going by secondary roads, instead of taking the highway.
I remember in particular the beauty of country-side Tuscany – and stopping in little creeks with nice landscapes whenever I wanted, instead of being conditioned to trains' schedules. With a car you can make many little detours – it is my favorite way to know the land and the people.

Train might be your easiest option, though there are coaches as well that are quite cheap! The coach might be a better option to get back up to Florence, money-wise. For a cheaper train ticket, get on one of the trains that make more frequent stops along the way. It might take a bit longer, but the countryside is gorgeous. I would not recommend renting a car in Italy unless you think you can handle Naples! My housemates are from there, both drive but avoid doing it in the city unless it's necessary. Also, there are loads of little commuter trains from Naples to Pompeii, so that should be no problem at all! And if you go on a Sunday, remember trains and buses don't run as often, so check ahead.