Day 5, I recommend you can take a ferry to Positano, Amalfi or Capri for the day, rather than exploring Sorrento. I really like Sorrento and have spent several days there. It is beautiful, but the other towns are prettier for exploring, IMO, and gorgeous arriving by ferry. I would pick Positano. Leave early. Have lunch in Positano. Take the ferry or bus back to Sorrento in time for dinner, shopping, and strolling around in the evening. Some people might prefer Capri. Either way, it would be a leisurely, lovely day, but you would still see a new place.

It looks like you will be paying to have a car for two full days that you are not using it, and only a short drive the first day. You might find that car rental, gas and tolls add up to more than the cost of trains and it is a lot more hassle, imo.

Whatever you do, you will have a wonderful time, and it will great to hear about your trip when you return.

Thank you all for your input! It's helped me to think through a few things and make a few adjustments that hopefully will make this a wonderful trip. This is a high school graduation present for our 18 year old daughter so naturally it has to be very special for her - we just get to tag along! I'll get you a report when we get back ;-)

When calculating the costs of driving vs. train, remember that CDW is required on all car rentals in Italy. Never mind what your credit card covers.

And you also need to get an International Driver's Permit to drive in Italy legally. But they're cheap and easily obtained at your local AAA office.

Pompeii and Vesuvius will take a whole day. The climb up Vesuvius is not too easy so keep that in mind. I worked out a lot before going over and doing it and it was still quite difficult. Pompei is HUGE. And I don't mean huge in normal terms, use any word that makes you think huger than huge and that is the size of Pompei. I think it would be v. difficult to just go through on your way to Rome. And def take the train into Naples, and then there are buses at the station or private hires which will drive you up in a van and are not too pricey. I took one as I didn't want to waste time waiting for the local buses.
Oh, and don't leave luggage in the car, it has a way of disappearing.
And if you have a chance def go to Positano - I stayed there for 7 days and didn't want to leave.
Hope you have a wonderful trip.