No. Slavery was a completely different concept to modern slavery of the type found in the United States before the Civil War. There was not really any concept that the slaves were any less a "person", although that entire idea is a bit too modern since citizenship was what really mattered. Slavery was often seen as a way of integrating subject peoples into the empire. Whenever the Romans won a battle or conquered a city the populace would be sold off into slavery. Educated Greek slaves for instance would often become tutors to wealthy Roman children, staff the civil service, be doctors and many other occupations. However, slavery was not a permanent state and an intelligent slave could often expect to be freed. Slaves would earn wages from their master with which they could buy their own freedom and that of their family and it was not uncommon for masters to manumit their favourite slaves in their will. After a slave became free he was called a freedman and while he still had many duties to his former master, like giving him a public escort and so on. despite not being full citizens many freedmen rose to high positions in society, most of Claudius' ministers were freedmen (Narcissus, Pallas, etc.) and Pliny the Elder claimed that many of them were richer than Crassus. Even so, the sons of a freedman would become citizens of Rome in their own right. While as far as i know Cicero had no servile relations i do know that he came from non Roman stock and by the time of his birth his family was already thoroughly integrated into Roman society. Other leading Romans were also from either non Roman elites or from directly servile stock at some point in the past, Trajan and Hadrian were both of Spanish origin and their family's wealth came from a brickworks.

That said, Slavery was normally no fun for the vast majority of those involved. Those in the countryside often had much less chance of being freed and often worked until they were too old to carry on. Similarly, being sent to the Spanish silver mines was essentially a death sentence for a slave. The treatment of slaves on Sicily was aslo especially harsh, as the numerous Servile wars fought there show. However, as we know, the Romans simply crushed the uprisings, never even considering abolition.

So in my opinion, there was no such thing as an abolition campaign because the cultural circumstance around which slavery and the institution of slavery were so different that it made very little sense for there to be an abolitionist movement at all.

The Roman Empire is not my area but I do study the antebellum and reconstruction South a lot and the slavery in either society was different. American slavery was based solely on race. There were a few exceptions but it was basically black=slave in the American South. Roman slavery was not race based and I don't think it had the chattel characteristics that American slavery had.

Roman history buff here, take what I say with a grain of salt. As far as I am aware, no. But that doesn't mean all was quiet on the issue. There were three major slave revolts throughout Roman history called "servile wars". The last and most famous being the Spartacus War. Yet of course the leaders of these revolts had no confirmed intention of wanting to end slavery, they mostly wanted freedom for themselves. But after the destructiveness of the Spartacus War, many slaveowners began to alter their behavior towards their slaves mostly out of fear. Being a cruel and bad slave owner was seen as being a cruel and bad pet owner today. The conditions, generally, for slaves became better over time. And as Christianity becomes more prevalent and Diocletian gets into power, you see the early stages of fuedalism and with that, serfdom which can be argued is a form of slavery within itself.