Still high season, so you can either adjust your plans to be where the people ain't or deal with the crowds. As you can tell, I don't like fighting the tourists, even if I am in fact one.
I am assuming you are flying in & out of Rome. I would not dissuade you from driving--especially that time of year--but please know what you are getting yourselves into through many informational threads on this forum. Me? I'd take the train to Florence immediately upon arrival at Roma Fiumicino airport, and base there.
For an agriturismo, that would be nice for 3-4 days, but if that's your base instead of Florence, you'll need a driver or a car. Daytrips with or without a car, you can do Fiesole, Lucca, Siena, Pisa, and/or any one of many hilltowns in the area. Then I'd take the train down to the Amalfi Coast, and base in Sorrento or Salerno, again 3-4 days.
We've ridden through Vico Equense and it looks beautiful, maybe the nicest views on that side of the peninsula but it's tiny and may not be quite the best base if you're active and want to get around, although the bus route & Circumvesuviana train do go through there.
Finally, take the train up to Rome and spend your last days there. Your decision as to whether you need to rent a car at any point in this journey.
Enjoy your planning!
