I really like polos and khakis-- I wear them to work every day. I know they're okay for guys, but can I wear them to dinner as well?
It really varies from line to line. On "mainstream" lines like Carnival and Norwegian Cruise Line, that outfit probably would be fine. On Celebrity, OTOH, the guidelines tend to be the minimum, especially for ladies, so you probably would be the most underdressed female in the dining room.
Also, will I be turned away if I'm wearing a denim skirt?
It would be best to leave the denim shirt at home unless you are going on a hike through a jungle or something of similar.
since you are going on Carnival - khaki's and a polo shirt with nice casual shoes or sandals will be just fine for casual nights, you might want to add a nice blouse or a twin-set just to mix things up a bit.
A jean skirt with a nice top, nice sandals or kitten heels is fine also for casual nights. I have worn a jean dress and sandals with no problems.
I just returned from a Princess Caribbean cruise last week...even with a dress code that was clearly stated "no jeans or shorts in dinning room on any night" there were plenty of both - even on formal nights...I actually overheard a family that was in sweats, jeans, and flip flops make rude comments about those dressed for formal while being seated......