i did a 28 day trip through sicily in june july entirely by public transport. Unlike what many people say, the roads were great and the busses and trains were pretty much on time.
what can i say, i loved sicily! sicily is italy but not quite. the influence of a couple of thousand years of disparate groups of people is palpable. you see and feel it, from the food and the architecture to the faces of sicilians, here are my thoughts based on your itinerary and my experience:
-palermo: it has so much to offer that 4 nights may be just enough. i am particularly fascinated by norman architecture and palermo and its surrounding has some of the finest examples. san cataldo, cappella palatina, la cuba and castello ziza to name just four in the city. Then there is monreale and the duomo in cefafu. both must see in my opinion! certainly agrigento is easier to get to from palermo by bus train or car. not to be missed-street food at the ballaro market!
-trapani: personally i didn't find trapani particularly interesting. i was there 3 nights but felt 2 would have been enough. good base for visiting segesta and erice though. segesta is less extensive than agrigento but the landscape is more picturesque. erice is a must-see with its serpentine cobblestoned streets. instantly takes you back hundreds of years. a day trip to mazara del vallo was a wonderful surprise especially the district called the kasbah with its whimsical wall decorations.
caltagirone: did a day trip. while the ceramics are amazing, the town itself not so much. the famous steps look better on camera. my opinion is based on only a few hours in the 'centro'.
ortigia: lovely with plenty to see. siracusa proper is less interesting but the ears of dionysius cave and caravaggio's 'the burial of saint lucy' (if you are into art) are worth seeing. yes, siracusa makes a great base for visiting the sicilian baroque towns. i don't think you can do them all in one day though. perhaps with a car. noto is my favorite but ragusa, scicli and modica are all beautiful, especially if you enjoy not just baroque, but sicilian baroque!
lipari: haven't been but i strongly suggest stromboli. been there several times and it's beautiful. the narrow streets, the sparkling white asymmetrical houses with colorful doors and windows, and of course the active volcano! stay overnight if you can. eat at punta lena and order the pasta alla strombolana. you won't be disappointed!
catania: great base for etna. acireale and randazzo are wonderful day trip destinations. especially randazzo with its unique black lave stone churches.
one place you have not included is piazza armanina. if you can, visit villa romana del casale. the mosaics are breathtaking. the city center is gorgeous too. i would not have bothered if the taxi driver (could be the much-recommended roberto) hadn't insisted i should see it.
i could go on and on. i love art, architecture and history so sicily was a dream destination for me. what i didn't expect was the experience of meeting the people of the island. i have to say they are some of the warmest and kindest i have met on my travels. they went out of the way to point me in the right direction when i could barely articulate my question. from my airbnb hosts in enna who insisted i have lunch with them every day to the teenagers in acireale who walked me to the bus stop, sicilians disproved over and over again every stereotype they are subjected to. anyway, hope this helps a little. enjoy your trip!