Three weeks in Sicily

I was in Sicily in spring of 2014. My very detailed trip report is here, and should give you some ideas to start: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

Three weeks is a lovely amount of time to have for Sicily. I urge you to pick up at least one comprehensive guide book to the island (get more if you go to the library rather than a book store) and read it. We don't know what might pique your interest. There's also a BestOfSicily website that is often recommended.

Be aware that Catania actually has more flights than Palermo. You might consider flying into one of those cities and out of the other to avoid the need to do a complete loop on the island.

There's a particularly nice group of Baroque towns in the southeastern part of the island. Do some reading on Ragusa, Noto, Modica, Scicli and others in that area. Ragusa has enough to see that I don't think two nights would be a mistake for the town itself (to allow for at least a full day). The others are smaller, and even via public transportation it is possible to visit two in one day, or perhaps three with some luck with the bus/train schedules. I visited from Siracusa so don't know how well Ragusa would work as a base for all of them.

I enjoyed a couple of day-trips into the center of the island, which is much less touristy than the coast. On my next visit I plan to park myself in the area and visit sites from a more convenient location rather than commuting in from Catania. Places to consider visiting include Enna, Piazza Armerina (Roman villa with super mosaics nearby, and Greek ruins at Morgantina), and Caltagirone.

Trapani and its medieval-hill-town satellite Erice are interesting places in the NW. Cefalu is now a touristy beach town but is a worthwhile and picturesque side-trip from Palermo.

There are lots of good Greek ruins, but that's not my thing. I did especially like the archaeological park in Siracusa. The city also has an archaeological museum that's good, to go along with the large medieval Ortygia district (my favorite such area in Sicily).

I was underwhelmed by the Egadi Islands, but I think the Aeolians are probably more interesting.

Having a car will make your trip more efficient, but you do not want to drive in Palermo! I'm not sure about Catania, either. If you start in one of the two big cities and end in the other, you may need to rent the car for only 15 days or so. You can get from Catania to Taormina relatively easily (or vice versa), and Taormina--built on the side of a hill--doesn't really need a car either.

Have you checked out RS’ Sicily Tour to see where he takes them? Also, if you go to that web site, you can check out comments made by folks who took that tour. We are about to sign up for that tour in May next year.

Good advice above. Fly into Palermo and out of Catania or vice versa. Rent a car after Palermo, turn it in when you get to Catania.

Palermo: a couple of easy daytrips by bus or train, Monreale and Cefalu.

Trapani: easy parking in big public lots, daytrips by car to Segesta and Erice.

Agrigento/Selinunte: temples, museum, nice town, good restaurants. Two nights here and you can drive to Selinunte, or stay in Selinunte Marinella and drive into Agrigento for a long day.

Piazza Armerina: beautiful, laid back city in the hills with Morgantina and Villa Romana del Casale a few kilometers away. The finds from Morgantina are in the museum at Aidone.

Siracusa: one of my favorite places. World-class archeological museum, archeological park; great restaurants in Ortigia, which has a big daily market and convenient parking.

Taormina: stunning setting, Greek theatre, beach. North of Catania.

Catania: you won't need a car here, there is a bus to the airport or a taxi for about €20. Stay in the Duomo/University area.

The Aeolian Islands can be reached from Milazzo via ferry; my favorite is Lipari, which makes a good base for seeing Stromboli, Panarea, Vulcano.

You can take day trips to the Baroque Towns(Noto, Ragusa, Modica) from Siracusa.

There is so much to see and enjoy. In September the beaches should be lovely for bathing too. (I was there in late April-early May and the water was too cold - but incredibly beautiful to look at.) I had 2 weeks and skipped a lot. I used 2 days to see Palermo, including Monreal. Then left most of my luggage at the hotel and spent 2 nights (1 day) in Cefalu by train with an overnight bag. I returned to Palermo to pick up my luggage and a rental car. Then I spent 3 nights in Castellammare del Golfo as a base to visit the coastal towns (Erice, etc) and the temples at Segesta and Selinunte. I prefer to drive more and change rooms less. From there, a day in Agrigento (leave time for the excellent museum) and spent the night in Piazza Amerina (wish I had 2 nights there, would have liked to explore the town more) and went to the superb Villa Romano. I skipped the towns of Ragusa and Noto (lack of time) and then spent several days in Syracuse, in the old town - Ortigia, and loved it; I used it as a base for Catania (excellent WWII museum) and Etna (on a day with poor visibility, so not worthwhile). With 3 weeks, you could add Taormina and islands.

I don't think it matters much if you start in Palermo or Catania. Fly into one and out of the other. It was easy to pick up a car in Palermo and drive out, though some people told me it was better to shlep out to the airport to avoid driving in the city. You definitely do not want a car there until you are ready to leave. I did drive into Catania for the WWII museum, the traffic was very slow, but not difficult to drive. I had no trouble driving in/out of Ortigia. I didn't get to Taormina bt I believe that's another place it's best to be without a car. I returned the car at Catania airport before my flight out.