Great tips above but I'd suggest at least one more day in Rome to hit up Ostia Antica as a day trip (
https://en.wikipedia.org/wiki/Ostia_Antica). It had amazing Roman ruins and was basically empty while we were there. If memory serves me right it's on Rome's train system and incredibly easy to get there. I also really enjoyed the Borghese Museum and saw some amazing sculptures there.
I'll also throw some love to Pisa. I had expected it to be a construction filled tourist trap but got pretty lucky while there with no construction and the Leaning Tower area was pretty awesome.
Hope you have a great trip!