Its difficult to say exactly how the Romans may have felt about any given ethnic group at any given time, but generally speaking, they were more concerned with culture and social standing. A Roman, and one of Latin descent, stood above all, whereas a barbaric appearing Gaul was considered contemptible. The poor were generally frowned upon as much as happens in any society, but Roman citizens were still more respected than any barbarian culture. Greeks were often accepted/admired (depending on the person or group of course) above various 'barbarian' cultures because of many similarities. However, evidence seems to show that a ethnicity played little part once that group was Romanized. If they looked, acted and behaved like a Roman, then to most, they were a Roman. Non Romanized foreigners were certainly subject to some disdain, though I'm sure.
Its difficult to trace the exact origin of racism as it relates to skin color, but we can assume it came in earnest with the age of exploration of the 15th to 18th centuries. Europeans found a vast land full of people that they considered to be different, primitive and beneath their own culture in general. The vast need for resources in a time of great wars and extensive colonization required the subjegation of these people to win the 'race'. As the subjegation of these people continued it certainly created an environment of superiority. Racism based on ethnicity (mainly concerning African people in this post) certainly existed prior to this point, but there were other factors involved such as religion, but its this age of discovery that truly began racism as we understand it today.