Welcome. I went on the Majesty for my first cruise 2 years ago and loved it! As far as excursions, it really depends on what you want to do. When you're in Nassau, I would definitely try to check out Atlantis. It shouldn't be more than $4.00 cab fare I wouldn't think. If you're into water sports, then I would recommend snorkeling or parasailing at either Nassau or Coco Cay.
For attire, during the day, shorts and a t-shirt (warm weather clothes) is ideal, and 2 bathing suits (so you always have a dry one). I think you will have 1 formal night in the main dining room. If you don't want to pack formal wear, they have several other options available, including the buffet. (Windjammer). Even though you're going on a short cruise, you don't really need 1 outfit for every day. I would just try to pack light and "mix and match." I know for some of us it's easier said than done. Here
is a link to Royal Caribbean's dress "policy." I put policy in quotes because some people and cruise lines have different meanings of "formal."