I will be in Naples (having already explored Naples/Amalfi Coast) with 5.5 days and 5 nights until I need to be in Rimini. I've looked into my options, and I think I want to visit Sicily. I have information on what to do while I'm there, but how do I get from Naples to Sicily (which cities, trains, ferries, etc.), and from Sicily up to Rimini?
Flying is the best option.
I agree, fly. I took Alitalia from Naples to Palermo a little over a year ago, it was €100 and flight time was an hour. You could fly into Catania if you will be on the eastern side of Sicily. Sicily is big and public transportation is slower than on the mainland.
Fly. Check out the lo-cost inter EU flights on Skyscanner.com. You can fly into Rimini, or Ancona and train, or Bologna and train. More and frequent trains out of Ancona.
Luke, As the others have mentioned, travel by air is the best option. From Naples to Sicily (Palermo or Catania), your best options will likely be Alitalia or Meridiana. For travel from Palermo to Rimini your best bet will likely be Volotea from either PMO or CTA to FLR (take the train from Florence to Rimini). However, note that they don't offer flights every day, so you may have to adjust your schedule accordingly. Flights directly to Rimini are probably possible (I didn't check), but they won't be as readily available. Happy travels!