Having gone on our first cruise a few summers ago with Royal Caribbean , I have come up with a few suggestions for booking and enjoying your first cruise:
Focus on the ship, not the destinations - The natural tendancy is to choose your cruise line by what ports will be visited. While this is important, I would say it is much more important to pick a cruise based on a ship that fits your needs. The reason is simple: you will spend the vast majority of your time on the ship, not at the ports. If you want a smaller ship for a more intimate feel, then do that. If you want to stay busy and want a ship with plenty of activities, look for a larger cruise ship. Also, make sure to know the age of your ship. Newer ships usually offer more ammenities than older ones, not to mention that they will be in better condition, and will have more up-to-date interiors. Cruise ships take a beating because of their heavy usage, so I would suggest you book a cruise on the newest ship that you can.
Make your port visits simple - You have to realize that one of the biggest limitations to many cruises is the limited time you have at each port. It takes a considerable time, because of the number of people involved, just to get off and on the ship. Plan your daytrips as simply as possible, and err on the side of underbooking your time rather than overbooking your activities. It is important to always keep a realistic expectation as to how much time you will have.
Don't get suckered into "adventure" plans offered by the cruise line - These packages for various activities at port destinations, such as snorkeling, scuba diving, ect., are usually very pricy. You have to realize that by booking these activities directly yourself, you can save significant cash. There is no magic to it, just do a little research and planning. If your needs are very basic, such as transportation to a beach, you can usually find transportation very easily when you get to port with no advance planning.
Take advantage of all that your ship has to offer - Because most of your time is spent on your ship, you should do all you can to enjoy and see as much of your ship as possible. There is a multitude of opportunities for eating, including the "Midnight Buffet" offered on most cruises. Check the schedule and try to hit as many of these activities as you can. Larger ships offer bands, dances, and other fun activites. The top decks usually have sports and outdoor features such as miniture golf, rock climbing, swimming, and basketball.
That about does it. And one other tid-bit that helped one of our party members out - if you suffer from sea-sickness, try eating a green apple. It really works!