In the deceptively simple-looking publication Exploring the NT World, Albert A. Bell, Jr. provides a useful discussion of slavery in the first century world (pp. 192-194). Bell supplies a number of primary sources to uphold his contentions, and what he concludes is that while first-century Christians did not push for an abolition of slavery, they did "call for an enlightened attitude toward slaves as fellow human beings" (Bell 194). Cf. Gal. 3:28.
Richard R. Melick, Jr. also has some perceptive observations in his commentary on Philippians, Colossians, and Philemon. (See pp. 316-320, 341-345 of his work.) Melick lists a number of good reasons why Paul did not protest slavery (railing against slavery could have made matters extremely difficult for those who were slaves!), while also showing that Paul "taught equality" and paved the way for first-century Roman society to make changes in its own thinking regarding slavery. In Colossians, Paul showed that those who were "masters" in a fleshly sense also had the Master over them (Col. 4:1).
Here is a more detailed passage from Melick's commentary on this issue:
"Slaves were, generally speaking, victims of war. The slavery was political and economic, not racial. Similarly, virtually every class of person lived with the realization that war could cause them to lose everything and be sold into slavery. Those who revolted, seeking to use power to gain freedom, found themselves in a worse position than before. It simply would not do for Paul to advocate slaves walking away from their masters. That would endanger many innocent lives and frustrate the spread of the gospel" (Melick 316).
Additionally, we read:
"Treating others the way they wanted to be treated [a message preached by both Jesus and Paul] would mean the release of slaves. As discussed earlier, however, the Roman Empire was not ready for that message, and to preach that kind of rebellion in those circumstances would have hindered the message of the gospel. The deeper matters of fair dealing had to come from the heart anyway, and Paul spoke directly to that end" (Ibid., 320).
Even if Paul or Peter had issued pronouncements against slavery, these locutions would have carried no weight with the Roman world, and later slave-owners would have found yet other reasons to perpetuate the institution of slavery and cruelly treat their human property. The fact of the matter is that neither the OT nor the NT encourages those who own slaves to treat them inhumanely (vide Exodus 21:20; 21:26). Even if God permitted slavery, he never urged masters to beat or mistreat their slaves: never would God have exhorted one man to hurl racial epithets at another man. Teachings against racism and nationalism are perspicuously contained in the early Christian documents.
The Apostle Peter said that God is not partial, and Paul reminded the Romans that people of the nations can worship Jehovah too(Acts 10:34, 35; Rom. 3:29; 1 Cor. 7:39; Tit.3:3). Furthermore, if one reads the entire account of Philemon, it quickly becomes evident that ancient Christians did not favor the enslavement of fellow humans(Philemon 13-16).
Exploring The NT World. Nashville, TN: Thomas Nelson, 1998.
Philippians, Colossians, Philemon. The New American Commentary. Nashville,
TN: Broadman, 1991.