Sea Princess has two restraurants, one has two sittings and the other is freedom or anytime dinning between 6 &10pm on formal nights it is customary to wear evening dress or tux in both these restaurants, as it is your first cruise you could use a dark loung suite or hire a dinner suit in case you don't like the experience of cruising.
There is also a Bistro type restaurant and a Pizza bar if you don't want to dress up but shorts etc are not allowed in the evening and slacks shirt and tie are the norm.
Sea Princess is a nice ship and you will anjoy it, have a nice time.
There are two formal nights per week, one semiformal and four casual nights. Because of a rather high number of UK guests, and the elegant reputation of Princess Cruises, about two-thirds of the guests wear tuxes on formal nights, but a suit is perfectly acceptable.
We have sailed on Sea Princess twice....I'm sure you will love it! My partner wears a black lounge suit with white shirt and bow tie for formal nights..and open necked shirt or polo shirt and trousers on casual nights. I usually wear a cocktail type dress for formal and black evening trousers with various tops for all the other evenings. You will usually be able to judge from the first evening what others are wearing. We saw some in Blazers and trousers on formal evenings so the rules aren't strictly enforced!!