As a certified diver, I would advise you to start before you go. That way you can experience more of the ship when you are at sea. During days at sea is when you do your training. Princess runs their program in conjunction with PADI, a very professional dive organization. They call it the New Waves program. Click the link below for more info. It would be best to sign up with Princess well before, that way they send you your books and you can read and complete the quizes before you go.
Any program that is any good will take hours of work and study to complete. This is much better and safer than a quick resort course. Diving is a wonderful and beautiful way to experience your vacation and done right, totally safe. Done wrong and it can be deadly. Remember anything worthwhile takes time and effort.