Hi there. Just off top of my head?
Fly into Rome airport - rent a car. Drive to Tuscany region.
Most of us do not advise this right off an international flight; jet lag can be a dangerous thing if unfamiliar with driving in Italy to begin with. Be sure and do your homework regarding unfamiliar signage and where you are and are not (ZTLs) allowed to drive. Stay in Rome on arrival day. As well, leaving anything at all you don't want stolen from an unattended vehicle is not advised. There's a "Definitive Guide for Driving in Italy" on this website (way down near bottom of the main page) that will be helpful:
https://italybeyondtheobvious.com/about/travel-philosophy/
Tuscany region: 3 days
3 days or 3 nights? 3 nights, if including arrival day, only gives you two full days to explore a big region.
Bari: 3 days
From Bologna/Modena? You're going to kill the lion's share of a day just getting there. Again, it doesn't give you a lot of time for a sizable area. And 3 nights/2 full days or 4 nights/3 full days? Honestly, with so little time I'd save this region for a different trip and add those days to Tuscany/ Rome.
Sorrento/Amalfi: 5 days
I would unload the car before your stay in Sorrento. Traffic on the coast can be a mess and parking is scarce and can be expensive. IMHO, use ferries, buses, commuter train or even hire a private driver for a day, although you'll likely get a range of opinions on the car-on-the-coast subject. There have also been restrictions on the days rentals can drive the coastal road during holiday periods based on odd/even license plates. I haven't seen yet what they'll decide to do this year but May 1 (Labor Day) is a big national holiday so it's very possible they'll enact similar limits around that time in 2024 to try and manage gridlock.
https://community.ricksteves.com/travel-forum/italy/return-of-odd-even-restrictions-driving-the-amalfi-coast
Drive Back to Rome for flight back to USA
It's usually advisable to stay in Rome the night before your flight.