Sicily

You will need to “work” the public transportation system.
Trains are good on the north and the northeast.
Trains not so good on the south and southeast.
Buses are …

We did Sicily last year all by public transit. We had 9 nights. We trained to Taormina. Had 3 nights there. Took bus from Taormina to train station ( a gorgeous station by the way!) and then train to Ortigia. ( 3 nights there). Then we took the bus to Palermo with 3 nights and took the bus to the airport as well.

There are sites that are not as accessible and with only the 9 nights we decided not to try to get to them. Agrigento and Raguasa we dropped, more due to time limitation as well.

I always say that you need at least two weeks to see most of the highlights in Sicily if you'll have a rental car for part of the time but need extra days if you're totally dependent on public transportation. You can get to most if not all of the key sights by train or bus eventually, but those are not fast forms of transportation, and you'll be dependent on the schedules. I spent 17 or 18 days in Sicily without a car, I skipped all of the large Greek sites (Agrigento, Selinunte and Segesta) and I still didn't get to all the interesting-sounding places I've seen recommended on this forum.

I'd say you cannot "do" Sicily in 8 days no matter what form of transportation you use; you can only see a part of the island. That may or may not be what you want to do; just don't try to cover the length and breadth of the island in that amount of time. Lisa has the right idea--when you only have 8 days for Sicily (even with a car), you have to be selective. To me it makes a lot of sense to focus on places that are comparatively easy to reach so you have more time for sightseeing and less time spent sitting in trains, buses and/or cars.

Palermo and Siracusa with the nearby Baroque towns are my two top choices because of the concentration of sights in those areas.

Some folks really like Taormina; other folks (including me) find it very touristy and don't want to spend much time there. It is undeniably pretty, but it will be a lot more pleasant if you can manage to be there when there's no monster cruise ship in port. Taormina could be a third stop if you don't feel you need to spend all your time in Palermo and Siracusa. If Taormina doesn't seem particularly appealing, you could make a stop in the interior between Palermo and Catania (en route to Siracusa).

Rick now has a guidebook for Sicily. I recommend getting a copy of it so you can see what the major options are and make your selections. He's very good about laying out logistical options, too. There's a high-level list of his suggested priorities (not going much below city/town level) on this very website. Follow this link and then click on At a Glance: https://www.ricksteves.com/europe/italy/sicily

I am not sure where Lisa was coming from that she took a train to Taormina. We took a bus from the Catania airport. Very easy.

We were among those who enjoyed Taormina. We were there in early May for three nights which was before the crowds. We also took a day trip to visit Mt. Etna from there.

Other alternatives available by public transportation are the Baroque towns of Noto, Ragusa and Modica from Siracusa. Cefalu is a quick train ride from Palermo

Even though public transportation is limited in Sicily, you still will have to make choices of what to see with only 8 days.

On a short trip, I'd base in Palermo and Catania and day trip from there. Both have lots of good food, culture and scenery. Even so, you'll be scratching the surface. After you prioritize your sights, you might find it best to just choose one and day trip from there. Eight full days is little time (I'm presuming it is; are you including day of arrival and day of departure?), and everyone is right, even for day trips, public transportation would eat up a significant amount of that precious time.

I liked Catania fine and later learned it has a WWII museum I would have liked to see. There's an attractive historic area. I'm not anti-Catania. But making that your east coast base probably takes the Baroque towns in the southeast (Ragusa, Noto, Modica and Scicli) off the table unless you have a rental car, because you'd first have to get to Siracusa and then scramble for a bus or train to your first (and possibly only) small town. Transportation to the small towns is not frequent. I liked all those towns, and I think they are nice contrasts to Siracusa/Palermo/Catania/Taormina. Even with a car, Siracusa and one of the towns would be more than I'd prefer to tackle in one day, so I don't think Catania makes a good base despite the car. It would work for me only if all I wanted to do in the southeast was take a day trip to Ortygia (the medieval part of Siracusa) and the archaeological museum and park. Ortygia is a fine area just for wandering around.

You may need to spend a night in Catania, whatever your sightseeing plans, because that's the largest airport in Sicily. A lot of folks fly into Palermo and out of Catania, or vice versa.