Sicily Tips for time and travel

I think you need to do some research first (i.e. read a book on Sicily and study the different regions, or see the below website to get you started) and figure out what you're interested in - then this forum will be helpful once you narrowed it down. Consider the effects of very high temperatures (and humidity) in late August/ early September.

http://www.bestofsicily.com/

You can't visit Sicily from one spot. It's too huge. You are going at a time that is hot, therefore staying along the coast, rather than the interior, will be cooler. Since you are there for a limited number of days, I suggest to rent a car, but not while in Palermo (or even Catania). So leave the two big cities at the end (or beginning) of your trip, so that you can stay there without a car. I don't consider Catania essential to visit, but Palermo (and Monreale on top of the hill) is a must. This is an example with arrival to PMO and return from CTA. You can do also the exact reverse.
WEST Sicily:
Palermo & Monreale: 2-3 nights. You could visit Cefalu by train/bus while in Palermo, but you might need an extra night.
Rent a car after Palermo.
Scopello. Use this as a base to visit the following: Zingaro Marine Reserve Park, Erice, San Vito lo Capo, Segesta temples, Selinunte temples. Wine region (Marsala). 3 nights at least.
SOUTH:
Drive along the southern coast. Stop in Agrigento to visit the Valley of the Temples. Also see the Staircase of the Turks (la scala dei Turchi), a unique cliff formation. You don't need to spend the night here if you don't want. Personally I would visit it on the go, then proceed to beautiful Piazza Armerina, where I would spend a night or two. While in Piazza Armerina, visit the Villa del Casale.
SOUTH EAST.
Syracuse and Noto are the must see places here. Spend a couple of nights or 3.
EAST
Taormina and Mt. Etna are the star attractions. 3 nights (2 full days) can suffice.
I think I already exceeded the nights, my friend. So I'm not even sure you have the time for all of the above.
Fly out of Catania (CTA).

While in Palermo stay in hotels. I don't think an apartment is worthwhile for short stays like that. Too much time wasted in the check in and out process.
While you have a car, besides countryside hotels, you could consider agriturismo (farmhouses). They provide good value for the money.
While visiting Syracuse I stayed at this one, between Syracuse and Noto. Beautiful beach on the premises:
http://www.masseriasulmare.it/

What Roberto said. I travel in Sicilia frequently, and it is a very big island. You can check viamichelin.com for driving estimates (including time and tolls, if any). With 12 nights, I would choose 3 base cities and daytrip from there. From Palermo, you can get to Cefalu, Monreale, Segesta.

We were there for just over a week over Christmas in 2013. The weather was mild and pleasant, although the one day at the Valley of the Temples outside Agrigento it was pretty warm . . . I can only imagine what a blast furnace it might be in the summer! Summer would give longer days, however, and it got dark pretty early in December.

Although we almost always prefer to stay in a place for a while, often taking "spoke" trips from our base (which we did in southern France this summer, but stayed within a relatively compact area), for our first quick visit to Sicily, we traveled light and did a bunch of one-night stands, mostly driving on to the next location, but stayed 2 nights each in 2 places. We didn't visit popular Palermo or Cefalu in December.

Highlights included ancient Greek and/or Roman sights at Segesta, Agrigento, Piazza Amerina, outside Caltagirone, and Trapani, plus strolling the beach near the Turkish Steps at sunset. We only stayed one night and part of a day in Modica, down in the southeast near Noto, but that was far and away the most satisfying place among a whole bunch of fabulous places in Sicily! I can't recommend it enough, and we will definitely stay longer in Modica on our next visit. It's a chocolate lover's paradise! Erice, high above Trapani in the northwest, was a delight, as well.

If you're interested, find an agriturismo to stay in for at least a night or two.

We rented a car, and in December the highways were a joy to drive - wide open, with courteous drivers and no tolls. If you drive (recommended), be prepared for very narrow streets in towns, and tight parking spaces. Rent the absolutely smallest car (with A/C) that will accommodate the 4 of you and allow you to safely & discretely stow your luggage.

Do you have any flexibility in the timing of your trip? We were in Sicily from mid-April to early May, and the temperatures were gorgeous but already getting warm. I have heard that August-September can be very hot, so if you can move the trip, it might be more enjoyable.

Temperatures are ok on the coast. Hot but often breezy. But as you move away from the water, it's hot. Segesta, Piazza Armerina, would be brutal.

I wrote a very detailed trip report about my Sicily visit in April-May 2014: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

As for flights, if you are not using miles as I was, the best connections from New York to Palermo or Catania are via Rome, Munich or Frankfurt. There are almost no nonstop flights from NYC to Naples, and if you connect in London, Paris, or Milan, you have to change airports. If you are making your way on separate tickets from mainland Europe to Sicily, Rome has by far the most daily flights, so that's the best choice.

Do plan on flying into Palermo and out of Catania (or vice versa), to minimize backtracking - unless, of course, you want to focus on one side of the island (and with only 12 nights, you certainly can do that).