We've been to Italy once and are going again this summer. I've researched tons on this topic and here's what we're doing...
For the Amalfi Coast we're going to base in Sorrento. We found an old monastery just outside of town with a private bath for about $100 euros per night. Rick Steves said Sorrento is a good base for this area too. I've heard too many bad things about crime in Naples and so am avoiding it. I think you would be better to cut off a day or two in the Amalfi area and add it to the Rome time.
Also, it would be cheaper to just take the car back to the rental agency and rent another when you're ready to leave a town, Rentals in Europe are expensive and for Rome and some of the other large areas you'll be in, paying for a car you won't be driving may not make the most sense. In the Sorrento/Amalfi Coast area the driving is notoriously awful and in the travle books I've been using tourists are warned not to drive in this area. I have more tips on sites if you need