Italy in 13 days

No offense, but you can't 'do' 'Italy' in 13 days. You may be able to see a good part of Sicily by car in 10 days with Rome for 3 but Sicily's public transport system might reduce your town/city accessibility.
Perhaps you can determine 3 or 4 towns of interest that we can help build on.....or decide whether Sicily or part of mainland Italy is the primary objective. We need more details to get started in the right direction for you.

Hi Maria,

(Do you mean you are leaving for Rome in 13 days or are you traveling within Italy/Sicily for 13 days total?)

Trenitalia - from Rome to Sicily - the carriages travel on the ferry across the Strait of Messina.
https://www.youtube.com/watch?v=9grLtdPWYhg

You need to start with a good guidebook, like Rick Steves’ Italy, to get an idea of an itinerary that will interest you. Why did you choose Rome and Sicily? Most people make Sicily a part of a trip only after seeing other parts of Italy. Sicily is difficult to see efficiently without a car unless you have LOTS of time, like a month,

Rick Steves’ guide will answer questions about getting around without a car and give you some basic travel skills for Europe and Italy. Reading his “Europe Through the Back Door” would be helpful as well. There is not substitute for reading and research unless you take an organized tour.

Just seeing Rome takes 5-6 days.
Just taking in Rome, Naples, Pompeii, Sorrento, Capri and the Amalfi Coast will take you 10 days, not sure you can fit in Sicily.
Before you go plan your trip in detail. Pick the specific cities that you want to see.
Don't plan on staying in any one place only one night, unless you can see it in half a day.
Factor in travel time, checking out of hotel or B&B, taxis to train station, train ride to next city, taxi to hotel, checking in hotel.

You need to start with a good guidebook, like Rick Steves’ Italy, to get an idea of an itinerary that will interest you?

There is no information on Sicily in the Rick Steves Italy guide books. Check out Lonely Planet. There’s a “Travel Tips” section on this website that gives a wealth of information that might help your planning. Posters here will be happy to provide guidance if you could give more specifics on what interests you, what you want to see and do. Remember that 13 nights = only 11 days.

Do you want to visit Rome for a few days then make your way to Sicily making stops along the way to visit Pompeii, the Amalfi Coast, Paestum? Or do you want to go straight to Sicily? Instead of taking valuable vacation time on the train (unless you want to take the uncomfortable night train) you could easily and economically fly to either Palermo or Catania. I know some travelers have used public transportation on Sicily but in the short time you have, outside of the major cities a car will allow maximum efficiency and allow you to discover and view the great sites the island has to offer.

EDIT: To give you a quick idea of what to see on the island, review the itinerary of Rick’s Best of Sicily tour. Or consider signing up. Of the 14 tours I’ve been on, it was my favorite.

There are cheap flights (well under $100) from Rome to Sicily. We're doing that in March, flying from Rome (Fiumincino) to Catania on Alitalia. I believe Ryan Air will also get you from Rome to Palermo or Catania, but they fly out of Ciampino airport. Getting from one airport to the other is not done easily. We chose to use Aliitalia so that we could simply fly from the US to Fiumincino and then connect there to Catania (3 hours between flights, so hoping for no big delay) without going into Rome until the end of the trip.

You can also take the overnight train (Trenitalia), but it sucks up a lot of your vacation time and you really don't get to see much.

It would be tough to imagine an itinerary without knowing much more about your preferences and such. I would think that since you haven't been to Italy before, you may want to devote 2 or 3 days at the start of the trip to Rome. Plus one full day at the end (sleeping in Rome that last night) before flying out. From Rome, you should probably fly to Palermo for the next 2 days. Then, without a car, you would need to stick to larger towns that can be reached by train or bus easily. Celfalu, to the east of Palermo is lovely. From there, you can go south to Agrigento in about 2 hours by train. West of Palermo is Trapani, which is right below the beautiful town of Erice. That might be a good destination to look into as well.