We spent an entire December in Italy as travelers before we moved there. In fact, we were in Italy from Dec 4-31. We selected 4 bases: Venice, Spello in Umbria, Sorrento, and Rome. Since you have 20 nights, you could do something similar that would give you a fabulous taste of Italy but not be rushing. This whole itinerary can be done by high-speed train.
Venice, 5 nights, Dec 17-21. On Dec 8 the Christmas lights come on all over Italy. Venice is magical and surprisingly pleasant in winter, although you will need warms coats, gloves, etc. Take a day trip to Padova.
Rome, 7 nights, Dec 22-28, over Christmas. I suggest a long stay because you will lose 1 1/2 days to holiday closures. But there's still plenty to occupy you even on Christmas Day. If you need ideas of what to do over Christmas in Rome, it has been discussed here on the Forum many times.
Florence, 4 nights Dec 29-Jan 1 for the museums, the restaurants, the shopping, the New Year's holiday. Take a day trip into the countryside, perhaps with Tours by Roberto.
Sorrento, 4 nights Jan 2-5 to visit Pompeii, Naples, the Amalfi Coast. Many Italians go here for the holidays so secure lodging as soon as you know you are going.