I usually go to trenitalia.com to repurchase tickets for train travel in Italy. The Freccia Rossa and Freccia Bianca are faster trains that make fewer stops.
Lets start from our arrival at FCO/Rome Fumicino. Once you leave the customs area and exit go to the right and follow the signs to the trains. Currently the train to Rome costs $14.00 or Euros and will get you there in 30 minutes or so. The bus is half the cost and twice the trouble, I recommend not taking the bus.
If you are making a multi-city trip and returning home from Rome it is best to end your visit in Rome as you will have to be there your last night anyway. You will lose a day if you park it in Rome for a few days when you arrive and then have to return again for a night. If you have half a brain you will find it easy to grasp this.
So lets assume your trip is two weeks long and you plan to visit Florence, Positano and Rome. I suggest that upon arrival in Rome you take the train to the station, then having pre-purchased your train ticket to Florence you go directly there. Remember your flight is going to arrive between 7AM and 9AM and you couldn't get in your hotel anyway. You should get your ticket to leave for Florence 2 1/2 hour after your arrival at FCO. Like you will get there earlier and might even make an earlier train but best be safe than sorry. You have two options if you arrive early: One try to change your ticket to an earlier train or my preference, just get on the earlier train and play dumb. If you have a ticket that you can change without fee it normally won't be a problem. I have done this several times, this is not AmTrak, the conductors are very nice and helpful. You will arrive in Florence in less than 2 hours.
There are several types of tickets you can purchase. I prefer business class and depending where I am going and when I will pay a little extra for an exchangeable ticket. The sooner you buy your tickets the more reasonable is the pricing.
Be careful at the train stations as there are gypsies who will attempt to help you with your tickets, bags, etc. They are really looking for your wallet. Be careful in all train stations and watch each other.
Pre-purchase your tickets because they will be cheaper and if you wait they will not be available when you want to go. The TrenItalia site is tough to navigate so be patient.
After a few days in Florence you can take the train to Naples and then get a driver to take you to Positano or whichever city on the Amalfi Coast. If you choose to go to Sorrento by train you will likely be changing trains in Naples and waiting etc. I would just call Josephine at italylimousines.it she will hook you up!