You'll be expected to dress for dinner each night at any of the dining rooms. Shorts are not acceptable for the dining room at dinner, but khaki pants are, and will be fine with a collared shirt. The kids might be able to get away with shorts if they're young.
If you're going on the 4-night cruise, some people dress up a little more when they eat at Triton's, but you could probably get by with khakis and a collared shirt. On the 7-night cruises, there is a formal night (Most will be in suits or tuxes. You'll need at least a jacket, at the bare minimum, for the guys) and a semi-formal night (could probably get away with khakis and collared shirt, but most folks we saw had at least a dress shirt and tie).
Luckily, Disney created a great option for folks who don't care for dressing up! Every night Topsiders serves the dinner menu from the main dining rooms in a casual environment. So, you can still have the nice food from the dining room, but you can dress however you feel comfortable. Also, there is room service all day and there are casual snacks up on the pool deck.
Hope you have fun!
