1 Timothy 1:8-10 - "We know that the law is good if one uses it properly. We also know that law is made not for the righteousbut for lawbreakers and rebels, the ungodly and sinful, the unholy and irreligious; for those who kill their fathers and mothers, for murderers, for adulterers and perverts, for slave traders and liars and perjurers..."
1 Corthinians 7:21 - "Were you a slave when you were claled? Don't let it trouble you - although if you can gain your freedom, do so."
Philemon :15 - "Perhaps the reason he was separated from you for a little while was that you might have him back for good - no longer as a slave, but better than a slave, as a dear brother."
The reason I say "now" is because slavery was allowed in the Old Testament. Jews were allowed to enslave pagans of other nations. They were not allowed to enslave a fellow Jew, one of the faithful. Believers were to be enslaved only to God.
There are some who believe that this was symbolic, like so much else in the OT, of man's hopelessness without salvation in Christ. Pagans, who were enslaved already with their pagan gods and enslaved by their pagan rulers, were enslaved by believers as a visible reminder of the curse they put on themselves.
Today, the covenant of Jesus Christ is for all nations everywhere. Therefore we have no idea who will be saved tomorrow and who won't be. Therefore slavery is unacceptable everywhere, because anyone can follow Christ, and should therefore be enslaved only to God. So the slavery practiced int he American South, which often involved Christian masters and Christian slaves, was especially an abomination to God. But any kind of slavery is offensive.
The base of the abolitionist movement was made up of a lot of Christians because they understood it to be an issue of the rights of man.
The 1 Timothy passage forbids slave trading. Yes it's an important distinction---slave trading often involves related kidnapping and violence. This passage is silent concerning a slave who voluntarily became one to repay a deb.
The 1 Corinthians passage simply acknowledges that being a free man is preferable to being a slave. But in fact, it clearly states that if you can't gain your freedom, you should "not let it trouble you".
Finally, the Philemon passage simply states the superiority of the Christian brotherhood relationship over the slave/master relationship.
These passages are even less relevant, then, than the various passages in the New Testament which command slaves to obey their masters and masters to treat their slaves with respect.
Indeed, the one specific prohibition of slavery in the Old Testament is that an Israelite was not to enslave another Israelite. Thus your later point, about not enslaving others because anyone can be a member of the new Israel, is a good one.