It is undeniable that all of Rome's achievements were underpinned by slavery. Hundreds and thousands of slaves were taken from throughout the empire to serve as tutors, construction workers and under Augustus even a public fire brigade. A slave could transition into a freedman/woman through manumission by their master or buying their freedom. At times the freedmen of the elite and emperors became very wealthy and influential. Take Felix for example who was a Greek freedman under Emperor Claudius and rose to the prestigious rank as procurator of Judaea in the 1st century CE.
However Roman slavery was of a different kind to the modern concept, it was not focused on a racial hierarchy but more a result of the socio-economic status of individuals. In fact the majority of slaves in Rome were of European descent - they were enslaved from Gaul, Germania and Greece. There was also slaves from Syria and Africa but these were firmly in the minority due to the long transit routes.
Ideas about race and hierarchy did not originate in antiquity, communities may have harboured prejudice and xenophobia against different groups but this was more a reflection of differing cultures, ideas and traditions rather than the colour of one's skin. By 193CE a person of colour had reached the top position of emperor: Septimius Severus. He was born in the African province of Leptis Magna - modern Libya and is a testament to the lack of importance colour had amongst the Romans. Severus was important in elevating further non-europeans to the emperorship through his marriage to a Syrian woman named Julia Domna. This created the Severan dynasty whose multiculturalism is evidenced through the elevation of local and Near Eastern culture in Rome. At one point, under Emperor Elagabalus of the Severans, Jupiter was replaced as the head of the Pantheon of gods with the Emesan (locality in Syria) god Elagabal. Even though this venture was short lived, it was not the colour of one's skin that caused issues but an attempt to change the traditions of the Romans which sparked friction.
The Severan Tondo c. 199CE. Emperor Septimius Severus and wife Julia Domna with brothers Caracalla and Geta (who has been subject to defacement) Reproduced from Wikicommons under Creative Commons.
I am not trying nor believe its possible to vindicate the Romans of deep prejudice, intolerance and xenophobia. These values are particularly prevalent in Roman attitudes to the Jews: in two conflicts (66-73CE and 132-135CE) the Romans burnt down the Jewish Temple, banned the Jews from entering Jerusalem and carried out numerous killings. There was a deliberate attempt to suppress the Jewish faith and this remains an inexcusable act of Roman imperialism.
Ultimately, it comes down to your definition of racism and if one's customs and cultural traditions can be seen as part of your race. However in terms of the colour of one's skin, the Romans did not subscribe to a racial hierarchy - that is a product of colonialism and the modern age.