You have two problems:
You are trying to tap additional power off a switch, which classically does/did not work. Because switches don't have power at them, since neutral is part of "power". Well, switches did not pre-2011.
The above problem wouldn't be a problem post-2011 when neutrals on switch loops became required. However your installation only has a /3 cable, forcing you to choose between a) having neutral on the switch loop OR b) having separate control of fan and light, which then forecloses any possibility of extending the circuit beyond the switch loop.
So if you use black for always-hot, and white for neutral, you only have red left over to control both fan and light. That only works if they switch together.
It sounds like you found it this way, and are now hoping to split control of fan and light without adding any wires. Well if it was possible with the wires in the wall, the last guy would have done that. Everyone likes separate control!
Option 1: automatic
Since always-hot and neutral are available at the fan box, maybe it's time for an automatic fan or a motion sensor light. Now you only need to switch 1 thing, and you have the wires for that.
Option 2: Replace /3 with /4 cable.
Better stores (read: NOT Home Depot or Lowes) will sell /4 or /2/2 cable by-the-foot. Now you have all the wires you need.
Option 3: Add a /2 cable to the /3.
This is tricky, because it flips the /3 cable from being a "switch loop" into being conventional "power at the switch" control of fan+light. The supply bypasses the fan altogether.
Supply hot+neutral comes into the fan box, and is immediately spliced to the new /2 cable and nothing else. Thus it is not tied to the fan in any way (except safety ground).
The new /2 cable enters the switch box as a power line. It splices into the onward /2 cable to other loads, and then, is wired as a conventional switch.
The /3 cable is flipped to be conventional - white is fan+light neutral...... black is fan, and red is light (or other way round, doesn't matter).