Expect luxurious hotels, beautiful setting overlooking the water, interesting ancient theater, prices that are higher than anywhere else in Sicily for everything (lodging, food, shopping items), tourist oriented pedestrian street full of shops, and a very compact place that you can easily see in one full day (maybe 1.5-2 days if you go to every single shop in a very leisurely manner). I don't know what it's like in December though, I went in September. You can wear anything you want unless some restaurant you're interested in has a very formal dress code. It's a beautiful place but wasn't my favorite in Sicily because I like more low-key places that aren't so geared toward (especially wealthy) tourists.
Are you going to Sicily only to see Taormina? If that's the case, I would recommend doing Sicily on a separate trip when you can see more of the island. Taormina is not like the rest of Sicily by a long-shot.