Question about Sicily

With or without a car?
My recommendation is a mix of public transportation (in the cities) and rental car (for the rural and smaller towns hard to reach by public transportation).
Palermo needs 4 nights, just to see Palermo and Monreale (located on a hill above Palermo). If you don’t have a car and plan to take day trips by train or bus in western Sicily, then you need to add one night for every day trip.
If you plan to rent a car to see things in West Sicily, then rent a car and get out of Palermo after you are done visiting the city.
South east Sicily needs 3 nights for Syracuse alone, but there are several places you can visit of day trips by bus or train, so add a night for each day trip.
Taormina needs 2 nights and so does Catania.
You have 4 weeks, so you need to decide what you want to see and determine the number of days in each base.
www.bestofsicily.com

No day trips. Just those 4 cities. Also not renting a car.

I think limiting yourself to four places is not the ideal approach. Sicily is not exactly huge, but it’s not small either. Like, sure you can drive coast to coast in something like 6 hours, but there’s a lot in between. Three of your bases are on the east side of the island, you’re basically leaving a full quarter of Sicily, the southeast which holds some its greatest sights, out of reach. With four weeks I would be basing in more places and allow for a more relaxed pace.

Of the places you mention: Catania isn’t on everyone’s tourist list, but those who have seem to report it’s little underrated. Still, it’s a truly urban city. The general consensus on Taormina is that it’s a bit of an overblown tourist destination, and many people prefer Ortigia (Siracusa.) Ortigia gets my vote for sure. Small, but effortlessly charming and lovely. It has real sights and makes a good base for that area. Palermo is one of my favorite underrated cities. Art, architecture and wandering aimlessly are its stock in trade. Terrific city worth some extended time.

I agree that you will be missing some fascinating places by limiting to those 4 cities. I really enjoyed the Valley of the Temples and Villa Romana del Casale.
Perhaps have a look at the AT A GLANCE section here:
https://www.ricksteves.com/europe/italy/sicily

Okay, let me change my parameters then. I still want to pick 4 places to see in a month, still no day trips and no car. But any 4 places. I assume Palermo and Ortigia would stay on the list, but what would replace Taormina and Catania?

I gave you the number of minimum nights for each place.
Taormina is very small and very popular and touristy, so crowded most of the year. It can be seen in less than a day. I recommended 2 nights so that you have a full day for it. But there are many places that one can enjoy nearby. I prefer to stay at Giardini Naxos, down the hill, because it has a beach and with a car it is easier to find a place with parking. Nearby you can visit Mt. Etna, Savoca, wineries (there are plenty in the slopes of Mt. Etna), also other towns like Acireale.
I don’t care for Catania much but there are nice things to keep you busy for a full day.
Syracuse/Ortigia has enough to keep you busy for at least 2 days (3 nights), but beyond that you need to venture out on day trips. There are plenty of beautiful baroque towns nearby (Noto, Modica, Ragusa Ibla), also a natural park (Vindicari). If you rule out day trips after the 3rd day you will run out of places to see in Syracuse.
There are plenty same is true for Palermo. In this case the city is bigger and has a lot to do. That is the only place that can keep you busy for a week. But Palermo is another place from which you can easily take a day trip to Cefalu, maybe Segesta temple as well.
But Sicily has a lot of great places outside of those major destinations, if you refuse to venture out on day trips or staying in other places, you are going to miss out a lot.

I appreciate everyone's responses, and I will take them to heart. But I do want to clarify something. We don't do 2 days here and 3 days there. We do 2 or 3 weeks in one place. We were in Seville for 3 weeks, no day trips. We were in Venice for 2 weeks, no day trips. We were in Barcelona for 2 weeks with one day trip to Montserrat. That's how we like to do it. So going 4 places in one month is probably pushing it for us, and while I appreciate suggestions like "You only need 3 days there..." that's not our style.

So I'm really just trying to get the backbones of a Sicily trip where we do move a bit more quickly than normal, which would be something like 4 places in a month. So far, definitely Palermo, and definitely Ortigia (Syracuse). And maybe rent a car and see something like that. But I just want to clarify the way we like to travel.

Why do you say you are looking for a replacement to Catania? A week there would be great; there may not be enough "sights" to fill all of your days, but it's a great city to wander around, soak up the atmosphere, and eat....

I would not spend a week in Taormina unless my intentions was to relax around a hotel pool for much of the time. (or spend time on the beach below town)

I think Ragusa would be great if you have the luxury of a week....you have Ibla,, with the narrow stone streets, and the upper town, laid out in a vague grid.....both very different in feel.....both enticing.

Might I/we ask: Why no day trips? Why you spend a week in Palermo without visiting Monreale?

I do get the plan to spend more time in one place...I tend to do that as well, but n most cases, not for an entire week. Example: Many people visit Cordoba, Spain, or a day trip or an overnight. I spent five nights there recently, and that gave me so much time to wander, explore out-of-the-way places, etc.... I love the idea of a week in Catania ro Palermo, maybe even Trapani, given your travel style....

I will change my mind and be open to a day trip.

Sorry, I guess I should have read up a bit more on Sicily before I started this conversation. My ignorance is obvious. I just suddenly got interested. We're headed to Spain in the Spring, and probably Germany in the fall, but I'm always looking for that next place.

Last October, we spent 5 nights in Palermo and I would have been happy to spend twice as long there. Not in the summer, though! Our first trip to Sicily was a couple weeks in July and that was way too much heat for us.

My husband wished for more time in Catania than the 3 nights we had there --- he thought the food market, Mercato del Carmine, was much more interesting than Palermo's heavily touristed one. A week there would suit you, I think.

Ragusa is where I would pick for a third week --- we were there 5 nights, staying in Ragusa Ibla, but actually Ragusa Superiore might be more your style. Ibla is very small and a bit tourist-oriented, although not in a bad way. You can walk or bus between the two parts of the city.

I don't know what to say about your 4th week --- I would sure hate to miss Agrigento and the Villa Romana. When we were staying in Ortigia for 3 nights, it was over-run with tour groups and school groups, so not our pick for a longer stay. I wonder if Trapani would be the kind of city you like to explore?

We hopped around a lot on both of our trips to Sicily --- not our preferred mode, either, but there is just so much to see and we figured it was our last trip to the region. Palermo is tempting me back, though.

I typically spend at least a week in major cities, plus extra days for day trips. Last year I had at least two weeks in both Amsterdam and Brussels and I'm about to take my third trip of 10+ days to Venice. I do understand your preferred pace, but I'd urge you to reconsider your no-day-trips position and see some of western Sicily. Rick has a guide book devoted to Sicily that will inform you of its many interesting sights.

With the exception of Palermo, Sicily--wonderful though it is--doesn't have a city with the breadth of sightseeing to be found in Venice, Barcelona or Seville. It does have--in addition to the places on your original list--three sites with magnificent Greek ruins (not near any of your original target bases) and a very interesting Roman villa with magnificent mosaics in the interior of the island. If you like exploring less touristy places, you might enjoy some of the interior towns like Piazza Armerina, Caltagirone and Enna. The northwest corner also has some attractive destinations. There are also smaller islands off the coast; I regret not having time for the Aeolian Islands.

To me, one of the joys of Siracusa is visiting the Baroque towns in that corner of the island--Ragusa, Noto, Modica and Scicli.

I think it would be a real shame to have a month in Sicily and miss the smaller places outside four (or five or six) base cities. Monreale, outside Palermo, is arguably the island's top sight, and the coastal town of Cefalu also makes a great (and easy) day trip from Palermo.

One logistical note: Sicily doesn't have a lot of trains. If you don't want to rent a car (and you definitely don't want one in Palermo), you will need to use buses sometimes. There are very, very few intercity buses running on Sundays and holidays.