How do I know if a ceiling fan with light and the switch are wired correctly?
From what you've described, this is what I think you're wiring should look like.  Keep in mind, this is just a guess. I can't see what's going on, and I'm an internet away, so I can't touch/measure/test anything.

The wiring diagram for the switch, can be found in the installation instructions.



If you provide images, and/or a better description of what wires and cables are in each box. I'll be able to provide a better guess, as to how this should be wired.
NOTE:
The diagram is based on your description, and is my best guess as to what your wiring looks like.  If you're not sure what to do, or feel like this project is over your head. Do not hesitate to contact a local licensed Electrician.