I was on the Explorer of the Seas the week before Christmas and the ship was decorated beautifully for the holidays (Christmas trees were in every nook and cranny - each decorated differently and quite unique). In talking to the crew about the difference between that week and the actual Christmas week - They indicated that passenger demographics was almost exclusively family groups; Christmas dinner menu was a little different than the normal menu; church services were more heavily attended; dress was slighly more formal (particularly children and ladies) on that day; but entertainment, daily activities, excursions were pretty much the same as usual. It is hard for the crew at that time of year because they do miss their families, and they DO appreciate the small gestures/gifts guests give - it does help their spirits!
Dress is very casual on the first night (Particularly since many people have not yet received their luggage, they must wear whatever they brought in a carry-on or generally - what they wore on the ship).
Weather is very unpredictable that time of year. It should be pleasant and mild, but cold fronts and rain are possibilities. (In that area of the Caribbean, at that time of year I have experienced sunny days in the 80's, mild misty rain, and one unforgettable cold snap that required us to wear sweatshirts/jackets wishing we had our gloves)
No matter what the weather, Enjoy the holiday on your first cruise -