Please recommend a trip to Sicily and Amalfi coast flying into Sicily and leaving from Naples.

I just did this trip last year. What I did was fly into Palermo to explore Sicily. From there, I took an overnight ferry to Naples. It was my first time doing this. It was sorta like a one night cruise (less the amenities) but it was cheap and got me there while I slept in a nice comfortable bed. Ferries only run certain nights and there are multiple lines with different schedules. From Naples, you are easy train rides to Sorrento and from there a bus or taxi to Positano. Hope you have fun!!

Thank you for the info. What cities did you visit, and how long did you stay in each?

Even if you were planning to rent a car for part of the time, you couldn't hit all of Sicily's highlights in 10 or 11 days. Since you want to to see the Sorrentine Peninsula as well, you're going to have to severely limit what part of Sicily you visit. I think the first thing you need to do is get a guide to Sicily (Rick now has one), read it and figure out what your top priorities are. Keep in mind that Sicily doesn't have the sort of fast trains you can take between major cities on the mainland, some destinations can only be reached by bus, and inter-city buses appear not to operate on Sundays.

I spent two days in Palermo, then three days between Sorrento, Positano and Paestum. Then I was off to Lisbon. I had been to Naples/Sorrento before so I visited Herculaneum and Casserta instead of Pompeii and Naples. I also spent a little time in Sorrento and then a full day in Positano and a half day in Paestum before flying to Lisbon. Positano is very lovely and I enjoyed that town a lot!! Palermo was nice but I probably would not go back. I enjoyed Herculaneum more than Pompeii and would absolutely recommend Casserta. The palace and gardens were amazing. I also thought Paestum was amazing and in some ways better than what I saw when I went to Athens.