Flight: $768
Trains: $90
Itinerary: Florence (4 days) > Rome (3 days) > Naples (3 days)
Trains: Florence > Rome 2h $60; Rome > Naples 1.25h $30
3.5 hours on trains
Hotels: 
We could have Project Expedition pick us up from the airport in Florence and have them show us the area for 3 days and 2 nights. They drop us off at the train station where we catch a train to Rome. 
When we are done in Rome, have Project Expedition pick us up and drive us south to Sorrento. There we see the Amalfi Coast, boat to Capri, take a tour of Pompeii, then they drop us back off in Rome! 
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(With Venice, would take about 12hours on train)
Flight: $804
Itinerary: Rome (3 days) > Naples (2 days) > Florence (2 days) > Venice (2 days) > Rome (1 day)
Trains: $150
Rome > Naples 1.25h, $15
Naples > Florence 3h, $35
Florence > Venice 2.25h, $25
Venice > Rome 4h, $40
Almost 12 hours total on trains. I wouldn't mind skipping Venice
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Itinerary: Florence (3 days) > Venice (2 days) > Rome (3 days) > Naples (2 days)
Trains: Florence> Venice 2.25h, $25; Venice > Rome 4h, $40; Rome > Naples 1.25h, $15
BUT - that is four different locations, living out of a suitcase by then.