The classic car tour of central Italy is usually small town rural Tuscany. There are many options but the classic southern towns of interest are places like Pienza, Montalcino, Montepulciano and San Quirico d'Orcia which are clustered pretty close together. There are also northern cities that can be toured in the same way.
This area is probably 3-4 hours north of Rome on the more rural road past Lago di Bolsena.
But if cars are an interest then centered around Modena in Emilia Romagna is Italy's "Motor Valley' are several places of interest including the Ferrari and Pagani Museums among others and also factory tours for other companies. These are often outlying places and difficult for people that aren't driving. I'm sure others will have additional ideas.
I think a balance of relaxed rural stays and sports car experiences sounds like an interesting mix. But I would also keep in mind that once you are in almost any major city in the north it will be faster - and I think less hassle - to drop the car and train into Venice instead. If you did visit the area around Modena I would look into dropping the car in Bologna and then taking the fast train into Venice itself.
Usual driving advice - read up on ZTLs, make sure every driver has an IDP before leaving the states, and add at least 25% to Google Maps drive times in the countryside.
Have a great trip,
=Tod