Sicily to Florence

Flying is the best option - either out of Palermo or Catania (it appears that Vueling flies direct from here).

Fly. Train would be about 12 hours.

Flights to Rome are more plentiful, so you might consider flying there and taking a train to Florence.

I flew on Alitalia from Catania to Florence with a change of planes in Rome. Cost was under $100 and my layover was at lunch hour, so it worked well for me. The new food stands at FCO are really good and not terribly expensive. I presume Alitalia service from Palermo would be similar. Veuling flights from Catania alternate between mid-morning and late afternoon depending on day of week, and they don't fly every day from Palermo. The Veuling flight was late afternoon the day I flew, which is why I opted for the morning Alitalia flight.

Edit: I considered the Pisa and Bologna flights, but for some reason they were considerably more expensive the day I was flying. Not sure if that's always the case.

Vueling flies direct non stop to Florence (from CTA and PMO)
Ryanair flies direct non stop to Pisa (also from CTA and PMO)
Alitalia (via Rome) has only 3 flights a day to FLR so sometimes it might make sense to proceed from Rome via train. Nevertheless I think Vueling is the best option if the times work for you.

Both Alitalia and Ryanair have flights to Bologna from both Palermo and Catania. That's another airport not far from Florence.

If you have a choice of Palermo or Catania, consider that Catania does have to shut down now and then when Etna spews ash.

I have taken the non-stop flight from Florence to Palermo on Vueling. The cost was < $100 and the service was good. I have not gone to Florence from Sicily, but flying from Florence to Palermo was easy.