A car is the right tool for the countryside and the wrong one for the cities. Rent it for the rural stretch of your trip and hand it back before the next big town.
Book an automatic well ahead if you do not drive a manual, because they are scarcer and pricier in Europe and sell out. Check whether your own card or travel insurance already covers the rental excess, since the desk will try hard to sell you cover you may not need. And carry your licence plus an International Driving Permit if your country needs one.
Pick the car up as you leave a city rather than at the airport on arrival, and you will skip a day or two of paying to park a car you are not using. Watch for the limited-traffic zones, the ZTL in Italy especially, that ring many old town centres. Drive in by mistake and the fine arrives months later by post.
Fill the tank before you return it, photograph the car all round at pickup and drop-off, and keep the drop-off receipt. None of it is hard, and a few days of driving the back roads, where the good villages hide, is worth the small admin.