Welcome to the world of cruising!
I second your insistence upon dinner in the dining room -- and this is especially true of the two formal nights, when the menu tends to be extraordinary! Since the "suggested" (that is, expected) attire for the evening applies not only to dinner, but also to all entertainment venues. Since some passengers have not understood this in the past, the Princess Patter on the first night of my last cruise clarified the matter with the heading, "Guidelines for Wear from 5:00 pm and thoroughout the Evening" above a summary of the eppropriate attire for casual and formal evenings.
My observation of gentlemen's attire on formal nights is a little different than Carole's -- more typically about even between dark suits and tuxedos or dinner jackets -- so your husband can go either way. It's also quite common to see a few Scottish gentlemen decked out in Scottish formalwear -- complete with kilt -- for formal nights on Princess cruises, and occasionally there are some folks wearing the formal dress of other cultures, too.
As to tips for first-time cruisers, I have three.
>> 1. Read the advance material thoroughly so you will know what to expect. This is especially true of Princess's Cruise AnswerBook.
>> 2. Think in terms of flexibility when selecting your wardrobe. For example, knit "golf" or "polo" or "tennis" shirts (with collars) are dressy enough to wear to dinner on casual nights, yet still casual enough to wear with shorts, slacks, or, for ladies, skirts during the day, either aboard ship or on excursions, or as a cover-up over swimming attire. Also, try to pack items that you can "mix and match" in different combinations. This will minimize the amount of luggage that you will have to shuffle through airports, customs, etc. en route.
>> 3. Don't commit the faux pas of referring to the ship as a "boat" -- especially in front of the captain. In the maritime world, a "boat" is a vessel that is not seaworthy. The insinuation is, of course, extremely insulting to the captain and the crew of any ship. Also, it would not hurt to learn a little maritime lingo -- port, starboard, fore, aft, deck, bulkhead, overhead, etc., -- before you go..
Other than that, just relax and enjoy a wonderful holiday!