How did the perception of slavery change throughout roman history and was there ever something like an Abolitionist Movement? Asking about the Roman Republic as well as the Empire.
The focus of my response will be on Christian points of view. Part of my post will look at Rome overall but it will mainly be focused on Christianity and the Late Roman Empire, especially the mid-fourth to early fifth centuries.
Short answer: no.
Long answer. No, there was not any significant debate about the morality of slavery in the Roman Empire. Slavery was vital to the Roman economy and political system. Although the number of slaves in the Empire fluctuated over time, estimates suggest that around thirty percent of Italy and ten percent of the rest of the Empire was slaves. Slaves would have been found in all sectors of the economy; from agriculture and mining to domestic housework and finance. Slavery was also incredibly common. Historian Kyle Harper (who wrote the book on Roman slavery FYI) estimates that those in the highest social classes would have owned hundreds of slaves. It wasn't uncommon for poor people to own slaves as well. Slaves were just another important asset to own, alongside land or livestock.
Now, on to the heart of your question. The Mediterranean gradually became more and more Christian throughout the first few centuries of the first millennium. Most Christians were fine with the practice and many owned slaves. There is no evidence to suggest that the average Christian owned fewer slaves or treated them better than any other Roman. The Bible mentions slaves many times and apostles interact with slaves at several points and no mention is made of slavery being negative or of the slaves being freed upon conversion. The apostle Paul writes about an escaped slave in Philemon and tells the master not to free the slave, but to not mistreat him when he returns.
The vast majority of Christians may have participated in, or at least been supportive, of the practice, there were Christians who spoke out against slavery. Augustine of Hippo (354-430) believed that slavery was a divine judgement from God onto humanity for the sins of Adam. But to Augustine, freedom was an innate quality that a person either possessed or did not, so slavery became wrong when free individuals were forcibly kidnapped and sold into slavery. In a letter sent to a bishop that was visiting Italy, he told of many cases where bandits kidnapped people in North Africa and transported them overseas where they would be disguised as "legitimate" slaves. And Augustine wasn't just complaining about this to a friend. He cited legal precedents and laws; imploring his friend to take up the issue with the emperor so action could be taken. But at the end of the day, he still supported slavery. In his letter he writes of an incident where members of his own church raided a ship that would be carrying these false slaves overseas.
Scarcely five or six were found to have been sold by their parents [note that it was legal for parents to sell their kids into slavery]; of all the others, hardly a person could keep himself from tears on hearing all the various ways by which they were brought to the Galatians by trickery or kidnapping.
The handful of individuals that were properly sold into slavery did not warrant any tears, but the others did.
To find a more definitive condemnation of slavery we must travel further east and several decades earlier. In a region of modern day Turkey, there lived three church officials that are now known as the Cappadocian Fathers. Two of them, Basil of Caesarea and Gregory of Nazianzus, critiqued aspects of slavery. They thought that slavery itself was fine, but one should take care not to own too many or else one would live too excessive of a lifestyle. Gregory of Nyssa (c. 335- c. 395) was different. In a sermon titled Homilies on Ecclesiastes he goes on what can only be classified as a tirade against slavery. It is remarkably similar to the type of language used by abolitionists in more recent history. Citing Genesis, he says that God did not give humanity ownership over other humans, only plants and animals.
Why do you go beyond what is subject to you and raise yourself up against the very species which is free, counting your own kind on a level with four-footed things and even footless things?
Humanity was made in the image of God, and God is free, therefore humans are free. To Gregory, freedom was an innate quality to all humans and to own another human in bondage was to go against God. Humans were slaves to God (it sounds bad but it just means to worship devotedly) but humans could not own slaves.
Perhaps the most rhetorically impressive portion of Gregory's rant is his section on the purchasing of a human.
For what price, tell me? What did you find in existence worth as much as this human nature? What price did you put on rationality? How many obols [currency] did you reckon the equivalent of the likeness of God?
The context of this admonition against slavery needs to be taken into account. Gregory is preaching a sermon about the sin of pride. In this instance, Gregory tells us that for a person to own a slave is for them to think they are at the same level of God.
Despite the harsh language, however, Gregory does not actually go so far as to call for the abolition of slavery. As mentioned above, slavery was a critical aspect of the Roman economy, so abolition would have been seen as absurd. His sermon was trying to influence their behavior, perhaps get his audience to treat their slaves better or even free them.
There was no such thing as abolitionism in the Roman Empire but that does not mean we do not hear voices of those decrying aspects of the brutal institution.
Sources
Augustine of Hippo. “Letter 10,” in Saint Augustine: Letters, translated by Robert B. Eno. Washington, D.C., The Catholic University of America Press, 1989.
Augustine of Hippo. “The City of God (Book IV).” New Advent. 2017.
Secondary Sources
Conley, Aaron D. “Augustine and Slavery: Freedom for the Free.” In Augustine and Social Justice, edited by Teresa Delgado, John Doody and Kim Paffenroth. New York: Lexington Books, 2015. 131-144.
De Wet, Chris L. "The Cappadocian Fathers on Slave Management." Studia Historiae Ecclesiasticae 39, no. 1 (2013): 1-7.
De Wet, Chris L. The Unbound God: Slavery and the Formation of Early Christian Thought. New York: Routledge, 2018.
Garnsey, Peter. Ideas of Slavery from Aristotle to Augustine. Cambridge: Cambridge University Press, 1996.
Glancy, Jennifer. “Slavery and the Rise of Christianity.” In The Cambridge World History of Slavery, edited by Keith Bradley and Paul Cartledge. New York: Cambridge University Press, 2011. 456-481.
Grey, Cam. “Slavery in the Late Roman World.” In The Cambridge World History of Slavery, edited by Keith Bradley and Paul Cartledge. New York: Cambridge University Press, 2011. 482-509.
Harper, Kyle. Slavery in the Late Roman World: AD 275-425. New York: Cambridge University Press, 2011.
Meredith, Anthony S.J. Gregory of Nyssa. New York: Routledge, 1999.
Here's a short video of Kyle Harper talking about Gregory of Nyssa. It covers this whole question really well.
I love talking about this so please ask follow-up questions if you have any. I'll do my best.
"Abolitionist," not that I know of, and "moral" is going to be a difficult term to pin down here, but certainly ancient authors discussed the institution, whether it was good or natural or vise versa.
Now, before we start, some basics about ancient slavery, (all of which can be found in K. R. Bradley, Slavery and Society at Rome (Cambridge 1994), or in just about any other Roman history textbook):
Slavery is much older than recorded Roman history, so we cannot sufficiently answer how it started.
In Rome, Anyone could be a slave. Unlike "modern" slavery, the ancient system was in no way race-based, primarily because the fraught modern concept of "race" didn't exist. See this Stanford Encyclopedia on "Race" for more. Now, this topic demands not only its own thread, but its own library. Suffice me to summarize, with regards to our discussion: ethnicity / skin color in the ancient world was not related to slavery.
Most slaves were prisoners of war; many were sold into slavery by their fathers, some sold themselves into slavery; others were forced into slavery as a legal punishment, or abducted by pirates and enslaved.
Slaves did all kinds of work. The worst lot was to work in the mines; the most fortunate were teachers or scribes. Cicero, a famous Roman orator/lawyer/rich guy, was on very good terms with his slave Tiro, whom he freed upon his death. Many gladiators were, of course, slaves -- it was not exactly a glamorous job!
Roman slaves had no legal personhood. They were the property of their masters. However!
Manumission (from Latin manus - hand + mitto to send, meaning to free a slave) was relatively common in Rome. In fact, Rome was full of freedmen, called liberti (singular male libertus, female liberta). The liberti had many citizenship rights (Roman society was highly hierarchical and many religio-political positions were still off-limits), and, importantly, could amass amazing amounts of wealth and popularity. The famous playwright Terence was a libertus as was Horace's dad (probably), as well as Trimalchio of Petronyius' Satyricon, one of the most famous characters in Roman literature. (Trivia: The Great Gatsby was originally to be called Trimalchio in West-Egg, as the titular character was based on Petronius' Trimalchio!) Upon manumission, one's former-owner would give the new libertus this cool hat.
This certainly did not mean all was peachy for ancient slaves. Many were treated very harshly or worked to death. Rome fought three Servile Wars (from servus, "slave"), the third most famously a slave revolt led by Spartacus.
Okay, on to the ancient sources:
The most famous ancient argument was Greek, not Roman, but many Romans (and thinkers throughout history) followed Aristotle's argument about "Natural Slavery" which he lays out in his Politics (and here is Jowett's 1885 translation, since it's free). Essentially, so it goes, some humans are fundamentally - by nature - worse than others. So, they are fit for slavery. Read 1.1.3-1.1.8 for a fuller discussion. Here's a bit pulled out, 1.1.5:
But is there any one thus intended by nature to be a slave, and for whom such a condition is expedient and right, or rather is not all slavery a violation of nature? There is no difficulty in answering this question, on grounds both of reason and of fact. For that some should rule, and others be ruled is a thing, not only necessary, but expedient; from the hour of their birth, some are marked out for subjection, others for rule. This argument should sound familiar, although warped: the justifications of modern slavery fused Aristotle's theory of natural slavery with Darwin's ideas of evolution to come up with the idea that some "races" had developed to servile to other, superior races. The modern development of racism is intimately related to modern slavery as an economic system, in sharp contrast to the ancient world.
Another must-read, this time an actual Roman, and much closer to a debate of morality, but about treatment of slaves rather than abolition: Seneca's letter to Lucilius 47, "On Master and Slave." He discusses that being a slave is basically just bad luck -- the fault of fortune -- and that owners should always treat their slaves well because their fortune could make them slaves at any time. From his opening:
"They are slaves," people declare. Nay, rather they are men. "Slaves!" No, comrades. "Slaves!" No, they are unpretentious friends. "Slaves!" No, they are our fellow-slaves, if one reflects that Fortune has equal rights over slaves and free men alike. ... (.10) Kindly remember that he whom you call your slave sprang from the same stock, is smiled upon by the same skies, and on equal terms with yourself breathes, lives, and dies. It is just as possible for you to see in him a free-born man as for him to see in you a slave.
So both writers are concerned with what makes one a slave, although neither here cited necessarily call the institution itself a "moral" issue. But I am no expert in ancient slavery (I focus on Latin poetry), and I know more sources exist! These are just a couple I remembered off-hand.
Here are three college syllabi for courses on ancient slavery, which you should poke through as you keep moving down this path of inquiry!
EDIT: With regard to your question: "How did the perception of slavery change throughout roman history," I'm afraid I am going to have to basically say "it's complicated." Any more response would be painfully reductive, so I am going to encourage you again to read some of the fantastic secondary source material on this! All the syllabi have biblios inside :D
I would also like to thank u/colorfulpony for his excellent answer
How were slaves treated in the Roman era (say Italy circa Julius Caesar)? Was the institution as awful as the more modern chattel slavery? Further, is there any record on how the slaves themselves thought about it?
I too would like to know this or perhaps point to a source that delves into this
Could you also say if the Kyle Harper book is good reading for a layperson?
There was some but slavery was generally considered just part of life. Anybody could be a slave. Romans were very class oriented and considered some people, like barbarians, were slave material. However unlike the Antebellum South there was never a detailed effort to declare that some people could ONLY be slaves because of the color of their skin or who their parents were or where they were born. If he shaved off his beard and talked like a civilized Roman and worshiped the right gods, and above all had money, a barbarian could be treated no differently than any other resident in the Empire. And unlike the South, slaves could buy their freedom or be freed by their masters and might even hope to get full citizenship some day if they were lucky and knew the right people. Unlike the South they still had a place in Roman society once they were freed. They didn't have to 'go north' to get respect.
Yes, great answer!! Just wondering about various aspects, if you’d like to comment further on any of these. Were slaves used in the military, e.g., as ‘cannon fodder’ (yeh, I know no cannons...), decoys, spies, etc? Could you be sentenced to be a slave forever or a period of years? Were war captives generally made into slaves?