Naples to Sicily

Rick has a guide to Sicily that will introduce you to a lot of the key destinations there.

For me, Palermo and the southeast (Siracusa and the Baroque towns of Ragusa, Noto, Modica and Scicli) were both key stops. The medieval section of Siracusa, Ortygia, is atmospheric, and there's a good archaeological museum in the modern city.

Catania is somewhat rough around the edges but has an attractive historic area around the cathedral and a WWII museum others have recommended. Some folks really like Taormina, which is very pretty but also very touristy (and expensive).

Cefalu is a popular coastal town east of Palermo.

I also enjoyed a visit to Erice, outside Trapani in the west, but I'm not very familiar with the island west of Palermo. Some earlier threads on this forum contain suggestions of things to see out that way.

I like to see island interiors since they are usually a lot less touristy than the coastlines. I'd recommend seeing the Villa Romana del Casale if it works with the rest of your itinerary. It wasn't crowded at all at the time of my afternoon visit during the summer of 2015, but others have had different experiences more recently. I suspect a lot depends on whether you hit the site around the same time as bus tours.

I skipped all the Greek-temple ruins, having no interest in them, and still spent 17-18 days in Sicily. There is a lot to see there. I did not rent a car, but a car would be very helpful for some parts of the island. Public transportation works pretty well for Palermo-Cefalu, Catania-Taormina and Siracusa. You can get to additional places by train or bus, but it will probably be slower. One thing to keep in mind is that there are very, very few inter-city buses on the roads on Sundays.

Before committing to renting a car on the mainland and taking it on the ferry to Sicily, check with your rental agency to be sure that is allowed.

Consider a flight unless there is a reason to drive. We flew Caterina to Naples and it was easy and fast. There is probably something into Palmero but the airport isn’t closed to the town.

We have taken the night ferry from Naples to Palermo and vice-versa (we used ferry hopper to find all the possible connections, here).
For Palermo, I suggest you go on maps and create your own itinerary (we based our own here, here and here). Since you are there, you could also organize a day trip to Cefalu for example. I hope that helps!

We took the train from Naples to Sicily. It goes on the ferry and then splits at Messina either to go towards Palermo or towards Catania. We got off at Taormina.

It was a long ride, but we enjoyed it. It was fun to watch them load the train onto the ferry and to be able to get off for a short while on the ferry as well.

In contrast our plane flight was delayed in Palermo and I think we spent longer that day travelling than we did with the train. But of course that was just some bad luck.
If you do take the train, they do not have a restaurant car. We picked up food ahead of time and in the station before.

I wouldn't hesitate to take the night ferry from Naples to Palermo. It's fun, you save a wasted travel day by transiting at night, and the value proposition is pretty friendly. Here is our experience from this past November:
https://community.ricksteves.com/travel-forum/italy/palermo-naples-overnight-ferry-experience

Renting a car on the mainland is not a great option. You'd have to pay to ferry it, you won't want one in Palermo and they have rental cars there when you are ready to leave.

I don't know how much time you are allocating for Sicily so bear with me here:
With 5 nights: Palermo 3 nights, Agrigento 1 full day but 2 nights. Sleep near the park rather than in Agrigento.
With 7: Add a pitstop at the Roman Villa del Casale to see the amazing mosaics then on to Ragusa/Baroque hill towns.
With 10: Either add Trapani 2 or add a night in Palermo and add a night in Hill towns
With 12: Add Etna (some people would rightfully argue for Ortigia), outbound through Catania
with more: Add Ortigia or Taormina?