This is not a direct answer to your question, but the modern Western conceptualization of "race" is a creation of 19th century positivist scientific thinking, and does not map well/at all on to the past. For example, when talking about pre-19th century Jewish persecution, one can talk about "anti-Judaism", but not "anti-semitism." The latter simply does not exist as a concept.

The Roman Empire encompasses a pretty huge stretch of time, and popular perceptions of foreign groups were often influenced by contemporary international relations. During late antiquity there were a series of laws passed aimed at suppressing the cultural practices of Germanic immigrants, who many Romans perceived as invaders. The Theodosian Code contains laws against practices such as wearing trousers, long hair for men, and certain types of shoes associated with the Goths. Additionally Goths were treated as outsiders in Roman towns during late antiquity. When Alaric began raiding Roman towns during the early 5th century some Romans reacted by organizing mobs and lynching prominent Gothic military officials. Pretty nasty stuff.

While certainly xenophobic in many instances, Roman attitudes can't really be called racism because the Romans did not have a concept of race. While black people in the Empire were perceived as exotic and sometimes mystical, they were not treated with the same sort of hostility that Germanic peoples were, possibly due to the friendly relations between the Roman Empire and the Kingdom of Meroë, where most black immigrants to the Empire came from. Some black people in the Empire were praised for their individual merits, such as Memnon who was an adopted son of prominent aristocrat Herodes Atticus. If you want more information on black people in the Roman Empire you should read any of the works of Frank Snowden.