Bumble Bee is quite correct about dressing "to the nines" on formal nights. On Princess, "formal night" does indeed mean FORMAL attire, and it extends throughout the ship. There's a Q&A booklet in one of your packets that explains the proper dress for formal and semi-formal evenings in more detail, and probably will answer many other questions that you might have. It's an excellent guide.
There are two things that I always do and highly recommend to others. The first is to spend some time familiarizing yourself with the ship's deck plans, and especially the locations of the various public rooms, before you go. The second is to take a walk through the ship to learn your way around as soon as possible after boarding. You will enjoy your cruise a lot more if you have some idea where to go for various activiites.
The "Western Caribbean" itinerary is still one of my favorites. Princess throws a great beach party on Princess Cays, so don't miss it! If you have never been to the other ports of call before, I do recommend an excursion as it's the best way to see as much as possible in a single day. In Ocho Rios, the tour of Prospect Plantation and Dunn's River Falls is an excellent choice -- and be sure to wear your swimsuit under your street clothes and to bring a towel from the ship and a pair of Surfwalkers (available at any sporting goods store) to climb the falls, which are well shaded and very refreshing. I suspect that any tour in Grand Cayman will be fascinating, but the "Sea and See" tour is an excellent choice for seeing as much as possible in a single day. Grand Cayman also offers the best shopping on the trip, with downtown George Town's duty-free shopping district right at the end of the pier -- and it's all high quality merchandise, too! Cozumel is a toss-up between the trip to Tulum (down side is that it's a long day -- and be sure to bring a big bottle of water for each person from the ship) or the much shorter tour which combines a visit to the Mayan ruins at San Gervasio with a Mexican Folkloric Show. The Mexican folkloric show on Cozumel is one of the best that I ever have seen.
Above all, relax and have a great cruise! For seven days, you will live in the manner to which you would like to become accustomed!