The Princess dress code is known as "smart casual".
For dining khakis & a polo are fine.
No shorts allowed in the dining rooms for dinner, but, certainly during the day.
If you care to have a casual meal you can always eat up in the Horizon Court Buffet.
Sabatini's & the Steakhouse usually follow the same drsss code, perhaps a bit more casual. You do not need to dress formally.
Hope you enjoy your cruise!