Sicily!

There are many places where relying on public transport is a good option. I'm not sure I'd say that Sicily is one of them. It would slow you down considerably and make routings inefficient and challenging. I'm sure it can be done, but the trade-off would be considerable. You might want to investigate what your plan - and budget - would look like using public transport versus a car rental...I think you may find that the trade-offs are significant.

I did not go to Agrigento, but I spent 18 days in Sicily in 2015 and traveled around solely by train and bus. You can easily get from Catania to Syracuse, you can see Noto and Ragusa (and Modica and Scicli) from Syracuse--though I wouldn't attempt the Noto-Ragusa combo on the same day since there's quite a lot to enjoy in Ragusa--and Trapani is linked to Palermo (far better by bus).

I had to look up the train schedule to check on Agrigento. I use the Deutsche Bahn for schedules, but if you're interested in fares, you'll want trenitalia. The latter requires that you use Italian city and station names (thus "Siracusa").

There's a 5-1/2 hour rail connection to Agrigento departing Syracuse at 1:35 PM. The earlier trains taking longer, slower routes. This one requires three changes, one in Catania (where you head inland) and two in Caltanisetta--a small city that unaccountably has two train stations. There's an even faster train leaving at 4:15 PM, but it follows that same pattern, requiring three transfers (not really a problem, usually, just sort of annoying).

You'll need to check on how you get to the Greek ruins in Agrigento from your the train or bus station or--more likely--from your hotel.

There's decent rail service between Agrigento and Palermo. But from Palermo to Trapani, a bus will come into play unless you just love interminable train trips.

You can start your exploration of bus service on Rome2Rio.com. Keep drilling down until you find a link to the bus company serving the route. Or you can try Googling "Bus CityA to CityB". Don't trust Rome2Rio's fares, frequencies, or travel times; go to the bus-company web site for the information you need. Confirm it once on the ground in the city from which you plan to depart.

Rome2Rio shows that you can, in fact, go straight from Agrigento to Trapani by taking the bus, or you can head out by bus and switch to a train at Castelvetrano, which takes a bit longer. I found the Sicilian buses comfortable.

Do not plan to go out of town on a bus on a Sunday or holiday. That way lies frustration.

It's possible that the train and bus schedules will be such that you'd find life easier if you visited the cities in a different order: Catania-Syracuse-Palermo-Agrigento-Trapani or Catania-Syracuse-Palermo-Trapani-Agrigento. You can get back to Palermo pretty readily from Agrigento, but it is more of a hassle from Trapani.

Fiddle around with the calendar first, given the probable need to avoid traveling between Trapani and Agrigento on a Sunday. If the bus looks faster than the train on another leg, you might need to avoid Sunday for a second travel day.

Then look at the actual times of buses/trains to see how much on-site sightseeing time you'll have. Without having done that, I can't say for sure, but at a glance your times look generally OK except that I think 3 nights is the bare minimum for Palermo unless you have almost no interest in the churches.

It would be nice to have a few extra nights so you could see Cefalu (from Palermo), Taormina (can be done from Catania), and/or spend a day in the interior (could be on the way from Syracuse to Agrigento or to Palermo) to see Piazza Armerina and the Villa Romana del Casale (mosaics).

I was quite lucky to take a guided tour of Sicily in summer 2017. We had a wonderful time at a very reasonable price. I normally go independent but quickly realized it is not so easy in Sicily. I took Gate one tours then added xtra time in Palermo At the end. I was very pleased with the tour, guide and all details.u might consider this as an option. Since public transport is not like mainland Italy. U will not be disappointed . Sicily is so underated. We stayed at a great B&b in central Palermo for about $ 70 For 2 persons.

I use public transportation in Sicilia all the time. You can have a great trip without a car. I also travel solo, on a budget. This is what I usually do:

I start in Palermo and end in Catania.

Cefalu is an hour by train from Palermo. Take a half-day to visit Monreale (bus from Palermo).

Bus to Trapani, daytrip to Erice.

Bus to Agtigento (spend a night or two, the temples and museum are worth the time). City bus will get you to the temples and to the museum.

I would spend a night (or two) in Piazza Armerina. There is limited bus transportation to the Villa Romana, but at worst you will have to spring for a taxi in one direction.

Bus to Siracusa (stay in Ortigia), daytrip to the nearby towns acraven mentioned (you don't need to see all of them). Siracusa has a world-class antiquities/archeological museum, in addition to many other sights.

Bus to Catania, daytrip to Taormina by bus. Fly out of Catania airport (you can see Mount Etna from the airport boarding area).

Post again if you want budget-friendly hotels/B&Bs.

Gail, we did almost your exact trip a couple of years ago.

We flew into Catania and went first to Taormina for a few days. Taormina is gorgeous; you might consider spending your first 2 nights there instead of in Catania ... it's a short trip.

We then took a bus from Taormina to Siracusa and stayed in Ortigia (one of our favorite places). We had considered a day trip from here to Noto but were enjoying exploring Siracusa & Ortigia, so we skipped it.

Then we took a bus to Agrigento. This was the longest leg of our trip ... we had to take a bus back to Catania, then another bus to Agrigento. My husband had caught a bad cold so he slept all the way, and slept an entire day in Agrigento, requiring us to change our plans slightly and add a day there. But he recovered enough to enjoy an outing to the Valley of the Temples (our B&B host had suggested using a driver from our B&B to the Temples rather than walking back to the bus station, which really worked out well ... he dropped us off and then returned about 3 hours later to return us to the B&B).

The next day we headed for Trapani, where we would immediately catch a ferry for 2 nights on Favignana. I woke up with the cold. We had planned to take the bus to Trapani but our host called the driver, who gave us a great price and drove us right from the B&B to the ferry. But this was just convenience (and luxury) for us ... the bus would have worked just fine.

We added another night to Favignana to allow me some recovery time. It was a quiet little island in early May. We rented bikes one day and rode to a gorgeous beach on flat, empty roads.

We took the ferry back to Trapani where now we only had 2 nights, and then got a free shuttle from our hotel to the Palermo airport.

We enjoyed Sicily so much, but there were many places on the island we still wanted to see. We were there 18 days and definitely not ready to leave. So we're going back in April 2018 ... again using public transportation ... flying into Catania, then visiting Acireale, Lipari, Palermo, Scopello and ending again in Trapani, flying home from Palermo.

I echo the advice to avoid travel on Sundays or holidays. Figuring out the bus schedules was a bit tricky in advance, but on weekdays buses run at least once daily between all of these places. So when we arrived in a city, one of our first tasks was a stop at a bus ticket office to figure out how to get to the next place. It was very easy to do. I know you will have a wonderful time in Sicily.

Here is a pretty useful site with lots of info on Sicily.
I used it a lot when planning my trip.....
Best of Sicily