This will be my first visit to Italy and I probably won't be back there for a long while, so I want to see as much as possible.
Day 2. In Florence (Evening train to Naples)
Day 3. Drive the Amalfi Coast and see Pompeii (Evening train to Rome)
Day 4 & 5. In Rome
FYI, when my wife & I spend a week visiting national parks in the US, we usually spend a day in one park and then drive at night to the next one. We are used to a rigorous travelling schedule and don't mind it all.