The earliest Christian Church seems not to have had the concept of heresy as we understand it. The term hairesis, from which we get our word heresy, referred simply to a sect or party. The Sadducees and Pharisees were heresies in that sense, parties within Judaism. But being a heresy was nothing necessarily negative.
On the other hand, the accusation against Paul as a ringleader of the hairesis of the Nazarenes (Acts 24:5) shows that the word could be used with something like the connotations it has for us. In spite of that, its full development into the idea of a set of non-orthodox beliefs or practices doesn't come until after the time of the New Testament. Heresy as we think of it is a post New Testament development.
But heresy in its ancient sense was a problem for the New Testament Church. The situation to which Paul is responding in 1 Corinthians says a great deal about that problem (and it also shows us an apostolic way of dealing with it). At the beginning of the letter Paul urges that there be no divisions (Greek: schismata) among them (1 Cor. 1:10), and he identifies what appear to be some of the parties: those who call themselves by the names of Paul, Apollos, Cephas, and Christ (1 Cor. 1:12).
The Book of Mormon shows us a similar problem, the division of its people into factions, and it identifies the reign of peace with the elimination of such factions, a time when "There were no robbers, nor murderers, neither were there Lamanites, nor any manner of –ites; but they were in one, the children of Christ, and heirs to the kingdom of God" (4 Ne 1:17). It is possible, in fact, to read the Book of Mormon as a whole as a demonstration of what happens to those who are unable to overcome factionalism.
Both the New Testament and the Book of Mormon understand factionalism to be evil.
Paul doesn't tell us the particulars of what the different parties in Corinth believe because his criticism is not of their beliefs. The fact that some identify themselves by his name suggests that he may agree with their beliefs, at least to some degree. But he includes them in his criticism because ultimately the problem is not what they believe so much as the fact that their differences have turned into factions within the congregation.
As Paul points out his preaching has not relied on "wisdom of words" but on the cross (1 Cor. 1:17). It isn't obvious what he means by "wisdom of words," though it is clear that the phrase is ironic: words alone will not give us the wisdom we need; that will come only through preaching the cross, the redemptive event of Christ's crucifixion and resurrection.
Yet words of wisdom are the very thing on which most often factions depend: those with ideas of one sort become one faction, and those with ideas of another sort become another faction, and each thinks itself better than the other. In fact, each assumes that the bad ideas of the other are unavoidably connected to disobedience of some kind, disaffection from the gospel. "We are those who worship most truly," each faction claims.
Paul does not understand the gospel in a way that leads to factions. In 1 Corinthians 2:1 and 2:4 he reiterates that his preaching did not depend on his superior powers of language or on him having a superior kind of learning or wisdom. Instead Paul's message is his testimony of Jesus Christ, a testimony through the witness of the Spirit and in power (1 Cor. 2:4).
The contrast with his teaching is "the wisdom of the wise" (1 Cor. 1:19), which is surely another way of talking about the wisdom of words. That is the wisdom which takes the wisdom of the world as its model rather than the wisdom of the cross. But the latter will look like foolishness to the world (1 Cor. 1:18).
Theological wisdom doesn't necessarily make the mistake of taking the wisdom of the world as its model for understanding, but when it creates divisions and factions like those against which Paul is preaching it implicitly does. Political wisdom carries the same danger. Perhaps every kind of worldly wisdom does.