As far as I know, there was no racism in the classical world, in the modern sense of prejudice by skin color.
I originally wrote this as a reply in comments to a previous post, but the subject deserves its own spot. The absence of evidence cannot be taken for evidence of absence, but it is the case that there is zero evidence for racism, not in Greece, and not in Roman times either to my knowledge. This might seem hard to believe for a modern reader, but anyone who wants to claim racism existed would need to come up with some solid evidence.
The classical world did of course have slavery, and lots of it, but this can't be equated with racism since they didn't care what color your skin was, and many societies were perfectly happy to enslave their own people and ethnically identical neighbors.
Social stratification based on skin color is not known anywhere in the classical world. I think the historical record is probably complete enough that we can say it either didn't exist, or if it did, was a pathologically small sample far from the norm. If anyone knows of a counter-example, feel free to tell us in comments.
The caste system was probably being invented in India around this time, and that probably counts, but India is a long way from the Mediterranean and is not normally considered part of the western classical heritage.
Tribalism however is very evident. The major conflicts in the Greek world are split between the Dorian and Ionian super-tribes. The alliances in the Peloponnesian War are split along Dorian/Ionian lines. But these tribes are genetically identical.
Similarly the Greeks and Persians had a tendency to kill each other, but this was clearly geopolitics and in particular a huge divide between the two in system of government; individuals married across the cultures and a number of high profile Greeks medized. (Medized means adopted Persian culture). Greeks who medized were looked on in contempt by other Greeks. This was because the Greeks considered themselves culturally superior to everyone except the Egyptians, so a Greek who medized was rejecting his own culture.
Other than Greece/Persia, the other great neverending conflicts of the classical world were Rome/Carthage and Rome/Asia. The Rome/Asia conflict was essentially a continuation of the Greece/Persia wars: geopolitics and culture clash and because, frankly, fighting each other is what empires did to pass the time back in those days.
But there was genuine repugnance between Rome and Carthage, the only instance I can think of where emotional hatred was at the core of an ancient war. The Romans were horrified that Carthage practiced large scale child sacrifice. The Carthaginians loathed Roman dominance. The Mediterranean simply wasn't large enough for the two of them. It led to three Punic Wars, which Rome was lucky to eventually win, and they razed Carthage to the ground to ensure there were no mistakes about a fourth war. It's hard to see this as racism though because, although Carthage is in North Africa, it was a Phoenician colony.