Naples, Sorrento and Sicily - how am I doing?

This sounds great to me since I have been to most if those places...just a remark...did you maybe think of including Mt. Etna on Sicily? This is really a great place!

I wouldn't try to include Mount Etna. The Sicilian part of the trip is pretty packed already.

I fiddled around with ViaMichelin to see whether the Villa Romana could be squeezed in, and I think it would be very difficult. Driving Siracusa - VR - Agrigento is estimated at 4 hr 3 min, and ViaMichelin is thought to be too optimistic. It takes a good bit of time to see the villa, and I assume you plan to do your Agrigento sightseeing that day as well. I suppose if you were willing to start your day at an ungodly hour, you could pull it off, but you'd probably pay the price the next day. Don't ignore the effect of the probably-hot weather you'll experience at outdoor sites like the VR and Agrigento.

If you instead try to insert the VR between Agrigento and Palermo, it means a longer driving day, and there's a lot to see in Palermo.

Be very, very careful about leaving things in your car when you stop for meals or sightseeing when you're in transit to a new hotel. Everything must be in the trunk before you arrive at your parking place. Costly and mission-critical items (electronics, prescription medications, etc.) should be in a purse/tote/daypack, not left in the trunk of the car. I don't think theft from cars is a greater risk in Sicily than it is in many other parts of Europe, but it is a risk (as it is in the US).

We were on the coast last summer (2017). Carpi was a disappointment (it was wall to wall people). I would suggest nixing Capri and doing Pompeii or if you are looking to be on the water, there's day trips by boat. We paid about $85/person for an all day boat trip (which is when we visited Capri). They took us to view the grottos and stopped for a swim, dropped us off on Capri for a few hours, then back to Positano with more sightseeing along the way. There's also boat trips that will take you to multiple towns along the coast on a single trip.

Thank you for the feedback!
@acraven: Our plans are to see the archeological site at Agrigento first thing after our overnight stay there. So perhaps there would be time to visit Villa Romano en route from Siracusa? My thought is it would be an interesting way to break up the drive, and a way to see a bit more of the interior of Sicily. Thanks for the reminder about visible luggage in the vehicle!
@hulagirl: I hear you about the crowds on Capri, but I'm hoping that early June won't be as bad as later in the season. What month were you there? My 17 year old is pretty enthusiastic about Capri, and we do have another day allotted for day tripping around the Amalfi coast, and a day to see Pompeii on the way back to Naples. Hopefully it will work to fit it all in, but I know we'll have to play it by ear somewhat once we arrive.

I would tweak your itinerary slightly. I would spend only 3 nights in Siracusa and then plan a night in Piazza Armerina to visit VR. On the way there, you could make stops in Modica and Ragusa. Then a night in Agrigento. From there I would drive to Monreale and then to Palermo for car drop off.

@Phillip That's an excellent idea!

Another thought - unless you are locked into flying out of Palermo - is to remain on the eastern side of Sicily and fly out of Catania. It seems that the bulk of places on your itinerary are in the East and adding Palermo cuts into your time. You could then see Villa Romana and the Baroque towns with comfort in your schedule.