You are probably not cruising somewhere chilly, which is quite rare since most cruises in January travel to destinations that are warm year round. Cruises to New England, Alaska and even Bermuda are on hiatus during the winter months.
Because of this, shorts, T-shirts and bathing suits will be quite comfortable. However, getting there might be a bit cold, even if you're leaving from Florida or California. Mornings and evenings may be uncomfortable, especially if it's windy. So for the beginning and end of your cruise when you are closer to the departure city than to your destination, bring along a light jacket or sweater and pair of slacks. Don't forget the socks and the shoes to go with them as flip-flops or sandals may find you in a nippy situation. You may also want to bring along a hat, which will not only help protect your head and face from the sun, but also keep your head warm if necessary.
Most cruises require casual dress for dinners. This means slacks and a short sleeve shirt, not shorts, tank tops, flip-flops and bathing suits. If you plan on attending a formal event or formal dining room, wear appropriate clothes. Depending on the cruise, it may mean a dark suit, tux or evening gown (e.g. Cunard Cruise). It might also only mean slacks and a polo shirt or a nice blouse and skirt (e.g. Carnival Cruise). Your best bet is to read all the literature provided before you pack to know exactly what their definition of "formal" is. Usually you can visit the cruise line's FAQs page and look for "What to wear?" as it is a common question asked by cruisers. This will answer any concerns over what to wear to dinner and formal events.
As mentioned above, don't forget the sweater or light jacket. You may even be chilly in the air-conditioned dining rooms, lounges and other indoor areas of the ship.
