You can wear shorts and t-shirts all day long although I did see a lot of people, even on formal night, in shorts and golf shirts wandering around the ship. However, shorts and t-shirts are not allowed in the dining room in the evening.........I also think that t-shirts with logos are not allowed in the dining room at any time of the day.......golf shirts yes, but not t's.
Well, I hope they would wear pants, whether they be long or short, any time of the day, except of course, when they are in their bathing suits.
. If you are at the pool, they ask that you wear a cover up and shoes/flip flops if you enter the Lido area in your bathing suit. Nothing like seeing some bare chested bloke standing in front of you in the Lido line while you get your food - UGH.
On formal nights, most men do keep their jackets and ties on, but you will always find those that go back to their cabin and change.
I can be hot in the Caribbean in January, but last week our first 3 days were quite cool and although we were in shorts, I had to put a sweater on even when I was outside. Weather is like a crap shoot - you never know what you are going to get.
Enjoy your first cruise. I am sure you will return for a 2nd one.