I always warn potential visitors that Sicily needs at least two weeks, and that's if you'll have a car for part of the time. Without a car, 17 days required me to omit places I'd have liked to see, and I had no interest in the Greek sights (Agrigento, Selinunte, Segesta), of which most people want to visit at least one. I love Sicily, but trying to plan a one-week visit there will be frustrating when you dig into the details of what is available.
You can get a better look at Puglia in a week, but it's long and skinny, so you'll need to consider carefully what towns you want to visit before selecting your base. Personally, I favor Lecce. There may still be some day trips offered from Lecce that take you in a van to multiple small towns and leave you to your own devices for sightseeing.
At the time you're proposing, I'd tend to head south for better protection against chilly, wet weather (which I heartily dislike), but also because it would allow you to see Rome, Naples, Puglia, etc., when they aren't perishingly hot. I spent a month in Rome (20 days), Naples (1 week) and Salerno (4 days) in 2022, beginning in mid-February. I had better weather than I had a right to expect, but your timing would be better (for me, at least), because the days would be longer and it probably wouldn't take until noon before the thermometer hit 40F. But it was great to see Pompeii, Herculaneum and Paestum in the afternoon without practically fainting from the heat.
If you're planning this trip for 2024, you'll be hitting Holy Week near the very beginning. Lots of people travel then. Big-name sights, of which there are quite a few in Rome, can get booked up. If you're interested in any of these places, check on ticket availability before firming up dates for visiting Rome:
Colosseum
Vatican Museums
Borghese Gallery
Domus Aurea
San Clemente (probably won't book up early, but get the ticket in advance)
In Naples, you need to get the ticket to the Capella Sansevero (for The Veiled Christ) well ahead of time; it definitely sells out.