I do love Berlin as well as Sicily. Berlin's easier, because you can basically stay put and just OD on museums. But if you're not up for weather that's likely to be rather cold and dreary, Sicily's an option. As always, advance planning is required to hit multiple small towns in one day (and don't even think about a side-trip by bus to a small town on Sunday), but there's some train service and buses fill in the gaps pretty well between the larger cities. It's true that you'd want to base yourself in at least two cities.
Depending on how you want to weight your trip--fabulous churches vs. general picturesqueness (is that a word?)--and exactly how many nights you'd have on the island, you might hit:
Palermo: at least 3 nights unless your tolerance for fab church and oratorio interiors is very low and you don't care about the museums. Monreale just outside town is a must. You can also visit the picturesque (but possibly dead in winter) coastal town of Cefalu from here.
Siracusa: at least 3 nights for Greek ruins, a very good museum, the large Ortygia medieval district filling a peninsula, and at least one side-trip. (Better more time for more side-trips.) Side-trips to the Baroque towns of Ragusa, Noto and Modica are possible; Ragusa probably needs a full day by itself. I assume those towns are still reachable by bus in mid-winter.
There's a very impressive Roman mosaic site in the interior that I was able to reach by public transportation in the summer. I don't know how viable that is in the winter. It's the Villa Romana del Casale, outside Piazza Armerina. There's a supposedly very lightly visited Greek site not too far away, Morgantina. But if you have a thing for Greek ruins, you'd want to include the SW for Agrigento and Segesta. I skipped them.
I liked Trapani on the west coast and the evocataive neighboring medieval hilltown of Erice. Many people prefer the lovely and beautifully situated Taormina on the east coast. It gets a lot of cruise traffice and was too touristy for me to stay long in mid-summer but should be much more peaceful in winter. It's your jumping-off point for visiting Mt. Etna. Taormina and Siracusa are about 2 hours apart by train. The major Catania airport lies roughly midway between them.
The trip between Palermo and Siracusa is via Catania and takes about 5 hours by train or bus, if one can believe Rome2Rio.com. That's not too bad if you don't have to return to your origin point.
Though you'd be hard-pressed to find someone less religious than I am, I really loved the religious sights in Palermo. If I had 8 nights in Sicily and I was flying into Palermo from Rome (no transatlantic jet-lag) and out of Catania, I might split my time 3 (Palermo) - 4 (Siracusa) - 1 (Catania, visiting Taormina that day). If flying into Catania and out of Palermo, I might go 1 (Catania, visiting Taormina on arrival day) - 3 (Siracusa) - 4 (Palermo).
