Much of the Christian pro-slavery argument focused on references to God's people practicing slavery in both the Old and New Testaments. In a society in which the Bible was one of the most widely owned and read books, arguments like these were powerful. The New Testament Epistles commend obedience to slaves, and the Old Testament does refer to several Israelites and patriarchs owning slaves. If slavery is bad, so the argument went, then why did the God of Israel not condemn it? The Letter to Philemon, for example, became especially popular as an apparent Biblical warrant for the Fugitive Slave Act, one of the most reviled pieces of legislation among abolitionists.
Anti-slavery Christian arguments took a few different forms. One was to argue that when the Greek and Hebrew texts refer to slaves and slavery, they are not discussing anything remotely similar to the kind of permanent, race-based chattel slavery practiced in the South. Slavery as practiced in the Bible, they argued, was not a permanent condition. Anti-slavery Christians also appealed to verses in the Old Testament condemning man-stealing, as well as injunctions by the prophets not to oppress the hireling in his wages. Kidnapping people and forcing them to work without pay, they argued, was against biblical teaching. Other arguments went beyond these specific verses to contend that it simply was not possible to live as a Christian and also keep slaves. The Bible may never have said "don't ever keep slaves," but the other things the Bible does command - to love others as yourself, to treat others as you would have yourself treated, and to seek justice - are not consistent with chattel slavery.
The 19th century produced several works of anti-slavery Christian literature, some of which are book-length, but you can also find copies of pamphlets that summarize some of these arguments. Charles Beecher's The God of the Bible Against Slavery is a good place to start, as is Does the Bible Sanction Slavery? published in 1836.
Secondary Readings
Molly Oshatz, Slavery and Sin: The Fight Against Slavery and the Rise of Liberal Protestantism
Snay and McKivigan, Religion and the Antebellum Debate Over Slavery
John Patrick Daly, When Slavery Was Called Freedom: Evangelicalism, Proslavery, and the Causes of the Civil War
Goodman and Sellers, Of One Blood: Abolitionism and the Origins of Racial Equality
Douglas Strong, Perfectionist Politics: Abolitionism and the Religious Tensions of American Politics