How did we get here?

Regarding the Greek influence, there is a connexion to Poseidon's trident potentially and also to Pan (as seen in this Roman mosaic) as inspiration for the "modern" depiction of the Devil that originate largely beginning in the 11th century.

Interestingly, to your question about him being red, one of the earliest surviving depictions that most scholars agree is a Christian depiction of the Devil is this 6th century mosaic from Ravenna. Christ is in purple, the good angel glowing in the ethereal light of heaven is red, and surprisingly the Devil is blue, as he is in the lower realm and different air. Also, note the goats as an indicative symbol.

The cosmology of Dante in the Inferno was highly influential in the late medieval and early modern depiction of the Devil. However, Dante was likely also influenced by artistic portrayals of the Devil that shortly predated him, like this Florentine baptistery mural, c.1260, by Coppo di Marcovaldo. Note the Devil is blue still but some of the modern characteristics are present. In the literary and visual representations in 13th century Florence we do see things like the wings and horns emerging.

The depiction evolves, like this early 16th century portrayal of Satan and demons by Luca Signorelli. This historical emphasis on the devil ruling/being condemned to Hell and the associating with fire and brimstone type of imagery is where the red colouring arises. The 16th century Christian Reformation that saw to the divide between Catholicism and Protestantism lended itself to an increasing emphasis on the persuasive power of the Devil on adherents of the opposing side.

Then enter the influence of depictions of the Devil as a seducer, think both like a Faustian pact or "deal with the Devil" and also in a sexual way like the fear of witches and witchcraft associated with the Salem Witch Trials. Out of this, the devil is a little less beastly often times and has a sometimes more appealing form. Some of these interpretations could also be bolstered by readings of Milton's portrayal of Satan.

Ultimately, this background leads to the amalgamation and where we get the red Devil in a suit like this modern painting of the lore of blues musician Robert Johnson at the crossroads, and the overly "cartoony" devil encountered by Jack Black and Kyle Glass in the opening "tribute." I've read lot of Augustine and some neo-platonic philosophy, I've also read Dante, Milton, and Goethe. At some point I had asked myself the same question of how we got the modern image we have of the Devil, as I just didn't have the entire context of the shift and looked into it. I hope this helps some as some starting points of inquiry. I decided to avoid a lot of the earlier, ancient and late antique Christian views on the nature of the Devil and just provide where the imagery started to form into what some of the current day depictions often portray.

Jefferey Burton Russel wrote a five-volume history that addresses the evovling perception and depicition: The Devil (1977), Satan (1981), Lucifer (1984), Mephistopheles (1986) and The Prince of Darkness (1988). It could be a good place to delve deep into this for anyone wondering.