So I'm trying to use a photo-resistor and I learned that you should have a "fixed" resistor in the circuit. So my question is, what resistor should I have be the fixed resistor? My photo-resistor has a resistance of 18-50 KΩ at 10lux and a darkness resistance of 2.0 MΩ at 0lux.
What part of the “light curve” are you interested in and what will you be connecting the LDR output / voltage divider to?
A photoresistor (LDR) that has a resistance of, say, 20k Ohms at 10 lux (which is a very low “candlelight” light level*) can have a resistance of only 100 Ohms in daylight (10.000 lux or more). Your voltage divider (input to an ADC i presume?) should be matched with the values you are interested in.
That said, if I make the assumption that you want to distinguish between daylight and darkness and you want to connect the LDR voltage divider to an ADC input on your Arduino that has the same range as your power supply (usually 5V or 3.3V), you could build the following voltage divider using your LDR and a 10k Ohm resistor (R1, your "fixed" resistor):
This will give you a fairly wide range of values (with higher values being brighter) and allows you (by experiment) to determine when it's time to turn the lights on, for example.
*Actually, the definition of 1 lux is the amount of light one candle generates at a meter distance, so 10 lux is the light of 10 candles.