If the moon was a bulb that you controlled with a light switch from your room, it would take about 2.38 seconds from the time you flipped the switch on for you to see the moon light up.
How did we get there? Well first, you need to know that light travels through space at about 186,000 miles per second.*
As it orbits, the moon’s distance from Earth is constantly changing. In January 2018, the moon got about 222,000 miles away, the closest it’s been since 1948.
From that distance, it will take 1.19 seconds (distance/time) for the moon to light up and another 1.19 seconds for that light to reach back to us here on Earth, for a total of 2.38 seconds.
*Please note that this is a hypothetical scenario that doesn't account for the electric current which is used to power on a light bulb. Electricity does travel at the speed of light, but electrons do not. So if there was a wire connecting the switch to the Moon, it would be slower, but we're using a little science fiction to make the point. :)