You seem to have an extra wire since red and black usually denote “hot”, wires. This would mean that the red/whit combination could be uses for one circuit, say, the light, while the black/white combination for the fan. However, you probably do not have 2 switches, and besides, the reasons you have that extra wire are numerous, and may not play well into what you want to do with it. One possibility is that the fan already has separate power sources for the fan and light, but the light switch controls only one of them. If you want to avoid adding a switch and any potential wiring hassles, I would suggest you go wireless.
It’s simple; the wireless kit is widely available at all the home stores. There are specific ones designed for a fan. The kit consists of a receiver, which mounts inside the fan/ceiling plate, and a transmitter, which mounts on the wall. In your case, simply remove your existing switch and connect the remaining wires together. Remember to turn off the circuit at the breaker box.
What you end up with is a circuit to the fan that is always on, or hot. This hot wire is attached to the appropriate receiver wire, then the remaining wires are hooked up to the fan wire, the light wire, and so on. Next, reinstall the fan to the ceiling, making sure the fan speed is on its highest setting, and the light switch on the fan is turned on. Last, place the transmitter in where the light switch used to be. The transmitter will control the 3 fan speeds, plus infinite dimming of the light(s). You will also need to read the kit’s instructions thoroughly, as there are other important things I did not cover, and variations between kit manufacturers.