In Rome, both in the city and the Empire, a lot of different cultures mixed, from northern Germanic to Sub-Saharan-African.
Where there tensions or problems based on a person's skin color or cultural heritage?
Where there segregation in time of war (family originating from Gaul but living in Rome facing backlash when there was a war or uprising in Gaul?)
Or where these things more or less ignored in favor of a person's (family's) social standing?
Racism per se is a modern invention (for a value of modern that means 'less than 500 years old'). Better to say that when the Greeks and Romans excluded, they had their own grounds and categorisations with which to exclude and other. More can always be said on the matter if anyone would like to toss in their knowledge; while we wait for new material, OP, this previous post by u/toldinstone goes into what the Greeks and Romans did exclude for.