What did the church of century one define as false teaching and false teachers?

Paul

As I urged you when I went into Macedonia, stay there in Ephesus so that you may command certain men not to teach false doctrines any longer nor to devote themselves to myths and endless genealogies. These promote controversies rather than God’s work—which is by faith. The goal of this command is love, which comes from a pure heart and a good conscience and a sincere faith. Some have wandered away from these and turned to meaningless talk. They want to be teachers of the law, but they do not know what they are talking about or what they so confidently affirm.
We know that the law is good if one uses it properly. We also know that law is made not for the righteous but for lawbreakers and rebels, the ungodly and sinful, the unholy and irreligious; for those who kill their fathers or mothers, for murderers, for adulterers and perverts, for slave traders and liars and perjurers—and for whatever else is contrary to the sound doctrine that conforms to the glorious gospel of the blessed God, which he entrusted to me. ~ 1 Timothy 1:3-11

I draw from those verses that false doctrines would include myths, endless genealogies which promote controversies – and teaching the law, which Paul in many other writings describes as supplanted by grace through faith. He continues in the same letter:

If anyone teaches false doctrines and does not agree to the sound instruction of our Lord Jesus Christ and to godly teaching, he is conceited and understands nothing. He has an unhealthy interest in controversies and quarrels about words that result in envy, strife, malicious talk, evil suspicions and constant friction between men of corrupt mind, who have been robbed of the truth and who think that godliness is a means to financial gain. ~ 1 Timothy 6:3-5

The primary interest of the false teacher here seems to be instigating quarrels about words, and making a buck by teaching.

This matter arose because some false brothers had infiltrated our ranks to spy on the freedom we have in Christ Jesus and to make us slaves. We did not give in to them for a moment, so that the truth of the gospel might remain with you. ~ Galatians 2:4-5

And here the heresy seems to be teaching circumcision as a corequisite for salvation.

Peter

But there were also false prophets among the people, just as there will be false teachers among you. They will secretly introduce destructive heresies, even denying the sovereign Lord who bought them—bringing swift destruction on themselves. Many will follow their shameful ways and will bring the way of truth into disrepute. In their greed these teachers will exploit you with stories they have made up. Their condemnation has long been hanging over them, and their destruction has not been sleeping. … This is especially true of those who follow the corrupt desire of the sinful nature and despise authority. Bold and arrogant, these men are not afraid to slander celestial beings; yet even angels, although they are stronger and more powerful, do not bring slanderous accusations against such beings in the presence of the Lord. But these men blaspheme in matters they do not understand. They are like brute beasts, creatures of instinct, born only to be caught and destroyed, and like beasts they too will perish. They will be paid back with harm for the harm they have done. Their idea of pleasure is to carouse in broad daylight. They are blots and blemishes, reveling in their pleasures while they feast with you. With eyes full of adultery, they never stop sinning; they seduce the unstable; they are experts in greed—an accursed brood! ~ 2 Peter 2:1-14

The false teachings Peter describes include made-up stories, destructive heresies – including denying the lord – and teaching by example that greed and adultery and carousing are perfectly all right.

John

Dear friends, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world. This is how you can recognize the Spirit of God: Every spirit that acknowledges that Jesus Christ has come in the flesh is from God, but every spirit that does not acknowledge Jesus is not from God. This is the spirit of the antichrist, which you have heard is coming and even now is already in the world. ~ John 4:1-4

The false teachings John describes are those which deny the incarnation or will not acknowledge Jesus.

Is there a pattern?

Aren’t these false teachings insisting on something besides the simple gospel of Christ as being just as important and necessary for saving the lost? Or even more important? Whether law, myths, made-up stories, or gnostic notions that the flesh cannot be affected by spiritual perfection – hence, that one can live carnally in the flesh yet still be spiritual; that Jesus could not have actually inhabited a fleshly body because it would have somehow corrupted Him – all of these are teachings which (instead of supplementing) actually deny the gospel. The gospel – the fact of Christ’s blood given, causing belief and penitence and change and transformation by rebirth – isn’t good enough. It isn’t powerful enough. It must be enhanced.

Now I ask you, are there any teachings today that approach that level of insistence on something else that is required for the salvation of a soul seeking God?

Something in one’s heart that must be believed?
Something in one’s life that must be achieved?
Something once done that cannot be forgiven?
Some sum in one’s purse that must be full-given?
Some man-made-up line that must not be crossed?
Some “holy” tradition that must not be lost?
Some interpretation that cannot be wrong?
Some kind of singing or some kind of song?
Something that excludes a gender or race?
Something that simply is false on its face?