Hi Vick, If you are flying into Palermo and only want to dedicate three or four nights in Sicily, then I suggest you use Palermo as your base and just explore the western side of the island. The city itself is vibrant with lots to see and there are excellent daytrips available from there. With just 3 or 4 days you just cannot see it all and will have to research what you really want to do. What appeals to you the most? If it were me, I would take all ten days in Sicily and limit my trip to Sicily and Rome, saving Amalfi and Naples for a future trip. Seems like you are trying to visit too many places in too few days but that might be the pace you like for traveling and that is fine. That decision is yours. Since you enter Sicily at Palermo will want to dedicate time to Palermo itself. Take a street food tour and visit the markets. If you like churches and religious sites there are plenty of those. Easy to walk around Palermo and take in the key sites. Lots of history there. From Palermo you can do daytrips to Monreale to see the beautiful mosaics and cloisters there, one day to visit lovely medieval mountaintop Erice with its stunning coastline views, regal Segesta with its Greek temple and Roman ruins, if wine is your thing maybe a visit to Marsala or area vineyards, with a stop at the Trapani area saltpans. if you like nature and beaches with those breathtaking views you want go to Castellammare del Golfo, Scopello, the Zingaro Nature Preserve, or San Vito lo Capo. And in the other direction there is pretty Cefalu. Another possible daytrio but one that most definitely needs needs a driver or a car to do it in one day, would be a visit to Selinunte ruins and nearby Mazara del Vallo. The Selinunte ruins are closer than the Agrigento ruins. It is about an hour and half car ride from Palermo. I very much enjoyed Selinunte archeological park. It is expansive, right on the coast with multiple temples and ruins and it just doesn't have the crowds that you will find at Agrigento. I liked Agrigento too, and Agrigento could also be done as a daytrip with a driver and I know it is the better known site, but I was there in late September a few years ago and it was very crowded while lesser known Selinunte was not crowded at all. Nearby Mazara del Vallo is a lovely fishing town whose historical center is a warren of narrow alleyways with lots of Tunisian influence.
I have not used a guide but did use a driver to take us out to Marsala, Erice and Segesta and on another day to other areas. Since you have such little time, a driver or driver/guide is very convenient to see what you want. If you want the name of the driver I used send me a DM. He tailored my two daytrips to what we wanted. The rest of my trip I relied on busses and trains. It is easy to get to Cefalu by either bus or train.
How will you get from Palermo to Amalfi Coast or Naples? Are you planning to go by ferry, train or plane? That decision will also impact what you can see and do.
If you go to Naples, consider a visit to Reggia di Caserta, The Royal Palace at Caserta.
