I would guess that the red wire from the existing switch ALSO goes to the light and fan, correct?

I have a good deal of experience with house wiring here in Canada, but the codes here tend to be more strict than those in many parts of the USA.

Up here, the main black/white supply lines typically pass through the "light outlet" (the ceiling box where the light and/or fan are mounted), and continue along to other outlets on the same circuit. A switch is wired in by running a wire pair (black+white, or black+red) from the ceiling outlet to the switch. When the switch is "on", the current just loops into the switch on one wire, and then back to the fixture (fan, light) on the other wire.

Cheers

-ml