Aside from your bathing suit, which will be your favorite uniform, take lightweight clothing (short-sleeve shirts, T-shirts, cotton slacks, lightweight jackets for evening wear for men; light dresses, shorts, and T-shirts for women). If you're going during high season, between mid-December and April, toss in a sweater for the occasional cool evening. Cover up in public places and downtown shopping expeditions, and save that skimpy bathing suit for the beach at your hotel.
Some of the more sophisticated hotels require jackets for men and dresses for women at dinner. The Bahamas' casinos do not have dress codes.