From my understanding/been told, the color red has the ability to travel the farthest in our atmosphere, being the reason why we have red sunsets and such. But with this in mind, why is it that at night the moonlight we receive seems to have a slight blue tint to it, and since red can travel the best out of our visible spectrum, how come our moonlit nights don't have a red tint instead of a blue one?
The blue tint is an illusion caused by the wavelength sensitivity shift when switching from rods to cones as the light intensity decreases below certain level.
It is called the Purkinje effect.
Objectively, moonlight is not blue. In fact, it is even more yellow than sunlight! (comparing the same elevation, because obviously the true color can be distorted by the atmoshpere, too). The moonlight is not bright enough to fully enable the photopic vision and our red color sensitivity suffers, producing the blue moonlight illusion.
There are a couple of slight misapprehensions here:
1) Red light doesn't travel furthest, it just is scattered least. From math.ucl.edu:
A clear cloudless day-time sky is blue because molecules in the air scatter blue light from the sun more than they scatter red light. When we look towards the sun at sunset, we see red and orange colours because the blue light has been scattered out and away from the line of sight.
2) In low light human eyes pretty much lose any colour sensitivity. Our eyes just respond to brightness. So while there may be a hint of a blue tint as no red will make its way round the earth - really it is just black and white and grey.