What order to o trip?

Fly into Florence, work your way south and depart from Naples, or the reverse.

Start in the north so the temperatures in the south won’t be as hot when you arrive.

Unless you're going for a month, the weather is likely to be about the same at one end of the trip as the other. Fly open-jaw into Rome and out of Naples. Start in Rome, then Florence, then Naples. Day trip from Naples to Pompeii by Circumvesuviana or Campania Express train (whichever comes first) and to Capri by ferry. I'd leave Naples for last because I wouldn't recommend it as anyone's first experience in Italy. Or fly round-trip Rome, go straight to Florence, then Naples, Rome last so you're near the airport for the flight home.

My big advice is to END in the city you fly home from. That saves backtracking and the common-but-unpleasant phenomenon of spending one pointless night near an airport instead of enjoying a destination. It's often easiest to fly home from Rome, in my experience (may vary by where you are coming from)

So maybe...
1. fly into Florence (or Rome and immediately take a train to Florence), visit Florence for however long you want
2. go to spend a few nights in Naples or Sorrento (easy base for both Pompeii and Capri)
3. go to Rome, visit it for however long you want, fly home from Rome

"It's often easiest to fly home from Rome, in my experience (may vary by where you are coming from)"

Yes, it very much depends on where you're starting from, so let us know. Rome has more nonstops from North America than other Italian cities, but depending on where you're starting from, you may have a nonstop to other cities like Milan, Venice, Naples, or Bologna (see below). Or, you may have no nonstops at all, which means you can start anywhere in Italy (if you have to make a change, you can change for Florence or Naples as easily as you can for Rome).

A great trick is that the Wikipedia page for each airport lists the nonstop flights from that airport.

-for the curious: nonstop flights from Philadelphia to Bologna (and to Naples), on American, start this summer! Since nonstops from JFK to Pisa stopped a few years ago, this will the the only nonstop service from the US to the part of Italy between Rome and Milan.

On another note, it's a bit complicated to spend a day in Pompeii and a day in Capri, and nothing else in that area. And while a day in Pompeii is very worthwhile, opinions are mixed about a day trip to Capri. Be sure you have enough of a reason to go there to make it worthwhile.

I also agree with Chani that Naples is simply not suitable as a first taste of Italy. I learned to like it, but it's much more "in your face" than the rest of the country (and yes, I've been to Palermo - no comparison). However, if you do fly into or out of Naples airport, you can take a direct bus to Sorrento, which is much easier to deal with than Naples, and has good transit links to both Capri and Pompeii.