We were in the Amalfi coast area the last few days of December 2012 and the first few days of January 2013. New Year's was fun watching all the fireworks go off around the hillsides. We stayed in Piano di Sorrento, next door to Sorrento proper, and used the Circumvesuviana for a day trip to Naples, took the ferry for a day trip to Capri, and drove to tour Ancient Roman sights including Pompeii and Herculaneumm, plus the drive along the Amalfi coast. Sights were generally open with generally regular hours, but as it got dark early, that limited what we could see outside in daylight. Pompeii ruins were supposed to be closed for New Year's Day, but wound being open with limited hours, so we were there with a fair number of other visitors (and an enormous Russian tour group). The water was too rough the day on Capri for the Blue Grotto, but had been accessible other days that week. Perhaps places were open thru Christmas and Epiphany, then maybe some start closing up right after January 6, but early winter had lots to see and do in the area. A local chef told us that April is when he would visit Sorrento, for the best combination of weather/sights/freedom from tourist hordes.
Temperatures were mild then, mid 50's Farenheit, but it occasionally rained for short periods, and one day (at Herculaneum) it absolutely poured. Locals carried umbrellas depending on the forecast (and most indoor sights in Rome 2 weeks earlier had umbrella checks inside the door, along with coat and bag checks), but it was never cold by Colorado or Minnesota standards.
The week earlier, Sicily had been a few degrees warmer, with rain on just one night in Taormina, but seeing snow-capped Mt. Etna from several vantage points around the island was awesome. We had a week with a car, and would love to go back again and see and do more. I understand in the summer it can be unbearably hot, so off season may be a good time to visit Sicily and southern Italy. The ancient Greek and Roman sights were fabulous, as was the food! We primarily used the Lonely Planet guidebook for great Sicily lodging recommendations on the go. Sicily highlights included Agrigento, Modica, and the hills in the Madonie National Park. Visiting some of Calabria on the way between Sicily and Sorrento was also wonderful.