Hi John D, welcome to Cruise-Chat. Waterart is correct, only the dining rooms are formal on formal nights. Princess also has a pizzeria that is casual attire, and don't forget there's room service as well. Depending on the ship you will be on, there may be other options available as well, some ships have a shushi bar. Once again, welcome aboard. Have a wonderful cruise!
Just bring along a jacket (no tie) for formal nights & you'll be fine. I bought the lightest jacket possible & have been using it for years + it takes up very little room in my small suitcase.