In Rome, near Vatican: VRBO 1021339, also a webpage (in Italian, but plenty to get the feel) at vaticanohouse.it Say hi to Laura for us. Her English is pretty good, and lots better than our Italian. Right next to Cipro metro station.
We found trains pretty inexpensive. Buy online in advance (nonrefundable); you get an e-ticket which you just show from your phone on the train. Keep an eye on the website for suddenly announced strikes (sciopero).
Don't rent a car in any Italian city. For several people, driving inter-city might be cheaper, but the driver will have no fun and will see nothing except fairly terrifying traffic. In a city, it's just an expensive, frustrating nuisance.
The Rome metro is cheap and fast, but there are really only two (maybe three now) lines, and so some areas where buses are better. Watch out for pickpockets. They will actually announce (electronically) on the metro when pickpockets have been sighted, and twice, locals announced in English "there are pickpockets in this car now," whereupon they left at the next stop. Often 2 or 3 kids, 10-13 years old, who stand in the doorway or create some commotion.
If you haven't seen
https://www.rometoolkit.com/ you should read it. Lots of great info.
IMHO the best site in Rome, after the Vatican, is Ostia Antica. In January, there will be practically no one there.