Italy

You need to clarify what you wish to see in Sicily first--it's a very large island with a lot to see. Most say allot two weeks for it, but of course that varies according to what you want to see. I would not do anything other than fly there and back.
1) Do you want to see any of Naples?
2) Do you want to stay in Sorrento and Positano or one or the other?
3) Do you wish to see Pompeii?
Only Naples can be called a city from what you have listed--Sorrento is a small resort town and Positano a village.
Once you flesh out how much time you need for the Naples area and on the Sicily, you can get into logistics.

Sicily needs more than two weeks. You need to chose either Sicily or the other, Naples, Positano, Sorrento, Capri( which merits more than a crowded day trip.) Either choice will be wonderful but combining them will not do either area justice.

In the event that you do decide to start in Naples and end up in Sicily, be certain to book your return flight from Sicily rather than from Naples, as otherwise you would be wasting a lot of time and money.

Pompeii is a must - even if you aren’t museum people

When we went to Sicily (our last trip before Covid), we went to Taormina, which is beautiful. We also drove around and spent a couple of days in different locations such as Agrigento, where we stayed in a agroturismo place. We also visited Siracusa, Trepani, castellamare del golfo, among others. Every place we visited was beautiful and different. Don’t go to Sicily unless you can travel around for at least two weeks.
Monica

Thank you all for the helpful tips but looks like I will have to re-think the plan as I obviously won't have enough time to do it all.

I do not want to see Naples and thought that was the best place to land to get around that area. As for Pompeii, I was there already on our last trip to Italy so I will not be going back there. If we decide to not go to Sicily and do Sorrento, Positano, etc. instead what is the better place to stay in of the two, which is less crowded and more budget friendly tho I understand that whole area is expensive.

Thank you.

If you are not interested in Naples or Pompeii, you will not need two weeks just for the immediate area. A few days on an island (Capri, Ischia, or Procida), and several days along the coast would be nice for a relaxed trip. Positano is not exactly budget friendly, but other villages on the Amalfi coast can be. Salerno is a more sedate, smaller city than Naples and many enjoy staying there.
What month will the trip occur?
If you don't want to fly to Sicily and back, you could head to Basilicata and Puglia. Or, you can take a train south along the coast.

Mostly likely September or October