My rough itinerary for 4 weeks in Italy. Comments , Advice please.

Get the new RS Sicily book. Lots of good ideas. The markets in Palermo were over the top. The one we enjoyed the most however was in Siracusa.

I am jealous! I stayed in a very nice B&B in Paestum called La Chora and recommend it. In Sicily, I loved staying in the small town of Piazza Romana near the Roman Villa. In you’re visiting Ragusa, stop at Modica and Scicli. Definitely base yourself on the island of Ortygia; great open air market and Casseficio Borderi for unbelievable sandwiches. I liked staying in Catania - great WW II Museum and a very large fish market. On one day, I drove up onto Mt. Etna and then circumnavigated the volcano stopping in small towns on the way.

I believe Philip meant to say he stayed in Piazza Armerina near the Villa Romana del Casale. I only passed through the town but it did look attractive. That interior area is interesting, with not very many foreign tourists. If you have extra time, you could check out Enna and Caltagirone as well. There are Greek ruins near Morgantina about which I know nothing except that few tourists are said to go there.

Other small towns near Siracusa and Ragusa are Noto, Modica and Scicli. They are interesting but don't need as much time as Ragusa, I don't think.

An interesting food fact I learned during my trip is that Sicilian pesto often includes almonds to smooth out the unusually strong Sicilian garlic.

As others have said, definitely Modica, Scicli and Noto, all of which I preferred to Ragusa, although that's worth a visit too. Every town in Sicily has its own specialities which means a slow itinerary is better.
Modica has a special type of chocolate for example,
https://slowtravelitalyspain.blogspot.com/2019/05/east-coast-sicily-day-10-modica.html?m=1

Scicli has the cake speciality called Testa di Turco but also the typical local Calzoni called Scaccia.
More info here:
https://slowtravelitalyspain.blogspot.com/2019/05/east-coast-sicily-day-13-scicli.html?m=1

If you are already planning on Mazara go to Marsala too where you can do tours and tastings of Marsala wine producers such as Florio.
Which month are you planning on going? There may be local festivals with an interesting food component.
Trapani is also a good base, there's also a good fish market every morning with local fish.
But the selection of fish in the small daily market in Ortigia Siracusa is even better. We usually stay in apartments so we can buy and cook our own fish and other food.
We found the centre of Palermo less interesting from that point of view. Ballero market is famous though and has a great atmosphere.

Also in Sicily ....Cannoli and Granite.... Italy Is great for food but in my opinion food in Sicily is the best in Italy

Palermo has three markets - Vucciria, Ballarò, and Capo. Our B&B owners recommended Capo (they said it was the only one they actually shopped at). Rick's new Sicily book favors Ballarò. You may want to check out all three to compare.

The market in Ortigia (the oldest part of Siracusa) is also very interesting.

Thank you everyone. You have help me a great deal. I will study, research and then make my final visit then. As far as the old Greek temples is concerned, I think I will visit Paestum, Agrigento and Segesta and that should be enough. Others will be food and churches.

I forgot to answer, I will be going in October , fly in and out of Naples. I probably do nothing in Naples since I am traveling alone and a female, too afraid. Would anyone have more advice for me to stay elsewhere instead of Vico Esquence? I already stayed there once and had been to the Amalfi Coast and Pompeii already. Where to go to see small town Italy( not Northern Italy ). Thanks.

Although Naples is large and hectic, I don't think you need to be fearful of it. I traveled there alone too, in my 20s. It has some wonderful sights to see that you won't see elsewhere (amazing museum, churches, Veiled Christ @ the Capella Sansevero, etc.), I wouldn't skip it personally. Plus some delicious food there as well, including sfoglietelle and baba cake desserts.

Have you considered dividing your time in Campania so that you're not just flying back to Naples to fly home? Maybe visit Paestum, then Sicily and then Vico Esquence. Unless your incoming flight arrives quite late, another thought would be to head directly to Paestum and save your day in Naples until the end of the trip.

In Palermo, I suggest visiting the Norman Palace when the Cappella Palatina is open. It has shorter hours than the palace, so plan accordingly.