We caught a train from Rome to Naples, then another train that you need to switch areas at the station for (but it is clearly marked) to Sorrento and then a bus to Positano.....long day and other tourists on the bus were very pushy!
On our return, we caught a ferry to Naples, taxi to the train station, and then a train to Rome, which seemed much more relaxed.
If you can get a train from Florence to Naples (which I haven't looked into), you can then get a taxi to the ferry and then catch the ferry to Positano, and it takes you right to the beach.
If you are willing to spend the money, you could also get a private driver.
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