Yep, knowing the ages your group is comprised of would be helpful. Knowing WHEN you're planning to do this would be helpful as well; for example, depending on what you want to do when you get there, the Sorrentine/Amalfi area isn't a great choice in winter.
Looks like there is train service into the Casentino Valley region from Arezzo up to Pratovecchio/Stia and some points in between. If wanting to stay in this area without vehicles, you'll want to be sure to book an accommodation IN a town and easy walking distance from a train station. Check locations on a map. Trying to move 10 people around at the same time will be a challenge so locations where people can head off on their own will be helpful. Seven days might also be much too long to stay this region, depending on everyone's interests.
The order that would probably make the most sense is Florence> Casentino Valley > Sorrento. Just off the top of my head, you'd take trains into the valley region from Firenze S.M. Novella to Arezzo and then regionale trains north into the valley villages/towns. Then a train back to Arezzo, train from Arezzo to Napoli, and then local transport (Circumvesuviana commuter train, hired driver, etc. ) to Sorrento.
If your flight from Naples is in the morning, you will want to stay in Naples the night before. Same for a flight from Rome.
(Edited a typo)