Should I rent a car in Italy?

For the countryside yes; for cities absolutely not. A car in Tuscany, Umbria, Sicily or Puglia is the difference between seeing the region and seeing its bus stops. A car in Rome, Florence, Venice or Naples is a parking bill, a ZTL fine and a permanent low-grade stress. The right pattern is to rent for the stretch you need and hand it back: train into Florence, pick up a car the morning you leave for Chianti, drop it in Siena before taking the train south. Rent from a city-edge or airport office rather than one inside a restricted zone.