France and Sicily help!

This sounds like a trip that will be a lot of train and ferry time and not much enjoying-the-destination time. Could you skip Nice and fly from Paris to Sicily?

You are going to be losing an awful lot of time getting from Nice to Naples, then to Palermo (ferry does not go to Taormina itself). Especially since it doesn't look like you particularly want to spend anytime in/near Naples itself.

Look into cheap flights from Nice to either Palermo or Catania. Alternatives could include a train from Nice to Turin, Genoa, or even Pisa, then a cheap flight.

As well as saving time, you will also be saving money.

Both Ryanair and BlueAir have cheap direct flights between Turin and Catania (airport nearest to Taormina). I have taken the mountain train between Nice and Turin (via Ventimiglia and Cuneo); it's a pleasant and very scenic ride (approx 4 hours).

Otherwise, there are also cheap flights from Nice to Rome, connecting onward to Catania.

I think Volotea (a Spanish low cost airline) plans to start flying from Nice to Palermo in 2016.

Currently there are no direct flights from Nice to Sicily (only to Naples, with Easyjet).

I would not travel from Nice to Naples via train, then from Naples to Sicily via ferry. You will lose 2 days getting there.

Are you going to Nice and Naples because you want to see these places, or because they are on the way between Paris and Taormina? I agree with the other replies, on a short trip such as this, you will want to fly between destinations; with advance booking, flights need not cost a lot either.

If you do want to see all four, the easiest way is to fly to Nice (nonstop flights on Delta from JFK). Then train or flight to Paris. From Paris, there are flights on various airlines to Catania, which is not only the closest airport to Taormina, but the largest in Sicily with the most flights (Palermo is second in size). There's a bus from Catania airport right to Taormina, so you don't need to go into Catania itself. Finally, fly from Sicily (either Catania or Palermo, depending on your Sicily itinerary) to Naples, then fly home from Naples (or, if you can't get the flight connections to New York that you want from Naples, spend a last night in Rome and fly home from there).

A trick I learned on this forum is that to find our where you can go nonstop from a specific airport, look at that airport's Wikipedia page. That's how I learned that Nice has no nonstops to Catania, but Paris does, on multiple carriers: https://en.wikipedia.org/wiki/Catania%E2%80%93Fontanarossa_Airport#Airlines_and_destinations

Finally, if you are booking a domestic Alitalia flight, use their "International" website rather than their US one. It often shows lower prices and more price options for the same flights. And book early - the cheapest seats are very limited in number and sell out fast. But to find all flights within Europe, a great resource is Skyscanner: http://www.skyscanner.com/