There are portions of OT and NT both which, when followed out, would logically lead to the abolition of slavery. St. Paul, to give one notable example in his letter to Philemon, expects Philemon to do the right thing and free Onesimus as a brother in Christ, without having to be commanded to be Paul (even though Paul says he can make it a command.) In the OT, to give another notable example, slaves are expected to be freed every Jubilee year: something that rarely if ever happened in practice, but the principle was there.
The Jubilee makes the best example, perhaps, because it illustrates something that either Chesterton or Lewis (I don’t recall which) once wrote: Christianity (and Judaism) could exist in a world with slavery. But it could also exist in a world without slavery, and looked forward to a time when this would be true; and to that author’s eye this was not something that could be said about any of the other worldviews on option at the time. Similarly, the beliefs could exist and operate in a world with war; but (like many other beliefs this time) could look forward to existing without war, too.
These things would be accomplished by the time (or by the coming) of the Day of the Lord. But it would be wrong to simply put off acting toward the fulfillment of these freedoms until that day. Thus the exhortation to peacemaking and slave-freeing in the Jubilee Years. Thus also the propensity of Christian cultures to sporadically attempt to end slavery in their civilizations throughout history.
The question of subordination of women is even more complicated; but it helps to be aware of cultural contexts, especially in comparison with other cultures of the time. In that regard, Judeo-Christianity was one of the more favorable environments for women to live and operate in by a significant margin. (Sociologist Rodney Stark, who wasn’t much interested in the religious truth-claims of Christianity at the time he initially started studying the factors, decided that the relative friendliness of the beliefs toward women was a key factor in the growth and maintenance of both beliefs in Greco-Roman society.) Was there room for improvement? Yes. But there was a significant amount of chivalry, too, when looked at from the proper contextual angle.
To pull a random example out of a hat: the ban against touching a woman during her period and soon after birth, helped protect her from infection during times when she was most vulnerable to blood-bourn illness. Circumcision is especially chivalric, as it leads to increased pleasure and safety for the woman, again helping prevent her from contracting disease. Yet I don’t recall these rationales being brought forth in the scriptures; perhaps because men simply weren’t in a position yet to understand and accept them as such. It’s one thing to cut off one’s foreskin to show how awesomely dedicated you are to faithfully following God. It’s quite another to do so to be so awesomely dedicated to serving your wife! (Especially when starting from a standpoint of almost no respect for women in a macho gung-ho culture of tribal raiders. From a hindsight perspective thousands of years later, what strikes me is that in setting up this symbol God effectively says that those men who aren’t sacrificing themselves for women are not being faithful to Him.) There’s an echo of this in the disciples plaintively asking Christ about His strictures regarding divorce, which when carefully examined turn out to be in favor of protecting women again: “If this is how things are to be between a man and a woman, then better not to marry!” Christ’s answer is pointedly sardonic–a fact often missed in interpretations: in effect He says that if men can’t live up to this, then the honorable thing to do is become a practicing eunuch!