Sicily & Southern Italy-Where to go & how much time?

I have not been to Sicily yet, but I know you can see just a corner or see it all--so anywhere from a few days to see one city to over a month to circuit the island. Most say two weeks is a minimum, so that's a good starting point. Regardless, just start reading to see what parts interest you.
I think a perfect Amalfi Coast+ itinerary is 9 days--a few in Naples if you are a history/city person, a few days on an island, and a few days on the Amalfi coast. Pompeii was a must for me, and if I had to do it over with more time, I'd include Herculaneum and more sites in Naples. Did you especially enjoy hiking in Cinque Terre? If so, I'd place more emphasis on the Amalfi Coast part because the hiking is similarly stunning.

Sicily needs at least two weeks. Then spend several nights in Naples and visit Pompeii. You can then stay on the Amalfi Coast or in nearby Sorrento. We spent five nights in Sorrento. Pompeii can also be visited easily from Sorrento as the commuter train stops right at the historic site’s entrance.
Sicily deserves the most time in your itinerary. The Greek and Roman antiquities there are incredible. You should rent a car when leaving Palermo, if that is your first stop.
You covered a lot in two weeks on your last trip to Italy so perhaps that is more your style. Most suggest two weeks for just Rome, Florence, Venice We prefer a slower pace and staying longer in each location. If you are still working, I understand. But if retired, slow down. Your largest expense and effort is in flying back and forth to Europe so stay longer once there. Sicily is not as expensive as other parts of Italy. We combined Sicily with Puglia , the heel of the boot on the Adriatic Coast.
We also fell in love with Italy and hope to visit again next year, our tenth trip there. We usually spend a month.

Here’s my response to a similar post. 2-3 weeks in Sicily will give you a good feel for the island. We had 2 weeks and a rental car on our trip. Places we stopped at included Catania, Ortigia, Noto, Villa Romana Casale, Agrigento, Selinunte, Marsala and the salt farm, Erice, Trapani and its salt farm, Monreale, Palermo, Caccamo, Cefalu, and Taormina. We skipped Segesta as it is much like Selinunte and we were templed out. We would have liked to see Modica and Ragusa, but didn’t have enough time. Our trip took place between late September and early October and the weather was great. Taormina is nice when there are no cruise ships in port. To me, one full day on Ortigia is sufficient unless you’re using it as a base to take day trips to other places. We didn’t drive in Catania or Palermo, rather, took a bus into the cities.

In 2014 we took a culinary tour with Earthbound Expeditions. It was called “Flavors of Southern Italy.” Rome>Amalfi Coast>Sicily. It was a 10 night tour. We considered it “a taste of the region.” We did add on to Rome (5 nights) Ortigia (5 nights) day trip to Noto, Ragusa, and Modica. Lastly, one night in Catania. It was our intention to return to Sicily. We were scheduled for 28 nights (Sept./Oct. 2020) starting in Rome, Palermo, Trapani, Agrigento, Ortigia, Milazzo, Lipari, Acireale, Catania. We planned on day trips along the journey. We were going to follow forum friend Priscilla’s itinerary using public transportation. (Thanks again Priscilla!)
https://community.ricksteves.com/travel-forum/trip-reports/my-trip-to-sicily-in-zoe-s-memory/
www.earthboundexpeditions.com

You don't say how many nights this trip will be. If two weeks including both travel days then you don't really have that much time for Sicily.
If possible fly into Naples and spend a night. Travel on to Sorento/Amalfi coast. We stayed in Sorrento as it was flat compared to the Amalfi coast, bigger and good ferry connections. There is a local train that goes from Sorrento to Pompeii. Visit Pompeii, Amalfi, Positano and Capri.
Then you can head to Sicily for 6/7 nights. You will have to decide which side of the island to visit, east or west . They are both wonderful. As far as the best places to visit, well the whole island. We were there for 18 nights June 2019 and we cannot wait to return. Taormina, Mount Etna, Ortigia/Siracusa Noto, Ragusa Palermo, Agrigento, Cefalu, Trapini.
you may have to do a little research to decide east or west.

On our Amalfi coast trip we stayed 3 nights in Sorrento which has a lovely cliffside setting on the Bay of Naples and spectacular sunsets. We took a ferry to Capri one day and enjoyed the Gardens of Augustus and Anacapri. Another day we took the train to Pompeii which was well worth it. Sorrento has a charming Centro Storico area of shops and restaurants. Getting a Sorrento hotel on the cliff facing the Mediterranean Sea is magical. We also stayed 3 nights in Positano on the Amalfi coast. It is not as easy to get to, but once we were there we took a ferry to Amalfi town and then an open air City Sightseeing bus to Ravello which has Villa Rufolo and Villa Cimbroni and great views. We only stayed one night in Naples to see the Archaeology Museum which was excellent. So 3 nights in Sorrento, 3 nights in Positano, and 1 night in Naples worked for us. We have not been to Sicily yet.

You can really fit any a trip to those two regions into any specific time allotment. It depends. If all you have is two weeks and really want to experience both then you could do it, you just have to realize you can't see all there is in that amount of time.

But you could pick a couple of places in each for 3 or 4 days each and have a nice enough trip.
If you have more time but still not unlimited a 'reasonable' amount of time would be 10-14 days for Sicily and 8-10 for the Amalfi Coast and Naples.
A pretty thorough exploration would need about 3 weeks Sicily and 2-3 weeks Amalfi/Naples.
And of course you could spend months in each. But that's a totally different type of travel.

I've only been to Sicily once. We had 11 days and felt that was a good amount of time to see the highlights without being rushed. I'd like to go back.

I've been to the Amalfi Coast multiple times. Our first trip was about 7 days and that was "OK" but I've been back several times to 're-see' a lot of it and to get to areas not able to be seen in just a week. Ideally for a first trip I would say two weeks.

Thank you, everyone. We will have a full 14 days in Italy. I don't need to see all of Sicily, just some of the best parts as well as some of the best parts of the Amalfi Coast/Naples. We are still working, so 3-4 weeks is not possible for us yet. It's great to keep that in mind when we are retired! Thanks again. You have all been very helpful.

I'd lean toward spending the two weeks in Sicily, but if you are very selective you can of course split it up if you prefer. There is just so much variety on Sicily, I see no reason to insert a flight or long train ride/drive into a trip, even with a luxurious two weeks!