Naples and Palermo

Best strategy is to peruse some of the threads here that include those places, consult a guidebook, check flight options, then post a tentative itinerary here with specific questions. Good luck!

PS When you say areas south, do you mean the immediate south of Naples (Amalfi Coast), or the entire coastline between Naples and Sicily?

We took the daytime train from Naples to Sicily. You could look into where that stops and see if that interests you. It is an interesting train ride as the train is actually loaded on to the ferry at the straits of Messina. Having said that it was most of one travel day to do that. Once the train reaches Sicily it splits to either go south to Taormina, Catania, Siracusa etc, or goes west to Palermo.
Sicily could easily absorb 14 days on its own, particularly if you are using public transit. It worked for us to do the other simply because we cannot get flights easily to Sicily and I had a cheap direct flight to Rome, but if you can get flights to Sicily I would leave Naples for another trip

Valdelphia has a great idea to check past posts.

When I was planning southern Italy + Sicily a year ago, we got some great info. See, https://community.ricksteves.com/travel-forum/italy/naples-and-environs-suggestions-per-favore and https://community.ricksteves.com/travel-forum/italy/southern-italy-itinerary-assistance-request. The first report has some amazing information from acraven that is Naples specific

We were persuaded to bump our plan from 3 weeks to 30 days because of comments that Sicily alone deserved 2 weeks. We ended up taking the RS 11-day Sicily tour, plus an extra day at each end; the extra day in Catania at the back end was not necessary. Our TR is at https://community.ricksteves.com/travel-forum/trip-reports/southern-italy-30-days-three-ways-to-travel.

Whatever you decide, Buon Viagge.