to teach or instruct: to hand on understanding so that the hearer grasps it and lives by it, and at times simply to tell someone what they are to do
This is the New Testament's ordinary word for teaching, and at its core it stays steady: someone who understands hands that understanding on so that others take hold of it. What is striking is not that the word changes shape from verse to verse but how much weight it came to carry. In ordinary Greek it was a workaday word for passing on a skill, a father having his son taught music or riding. The Greek Old Testament put the same word in a new place, using it above all for God teaching his people his ways, and for parents, priests, and scribes handing on that instruction from one generation to the next. By the time of the New Testament the word had picked up that whole background.
Most often the word carries this full sense of teaching. Now and then it means something plainer, to tell or direct someone what to do, as when soldiers did as they were instructed, or when someone leads others into a course of action. The one word covers both giving directions and handing on understanding.
So when the Gospels use it constantly for what Jesus did, often pairing it with proclaiming and healing, they mean more than a lecture. His hearers noticed that he taught as one who had real authority in himself, not as someone merely repeating received opinions. The letters use the same word for the steady instruction that forms and builds up a congregation, and for the handing on of a fixed body of trusted teaching from one faithful person to the next.
The word itself passes no verdict on what is taught: the New Testament can use it just as plainly for teaching that leads people astray. That is why teaching is treated as a serious trust and false teaching as a real danger. And running underneath all of it is a claim the New Testament makes its own: that behind every true human teacher stands God as the real teacher, so that the Spirit is said to teach, and the promise that God's people will all be taught by God is taken as coming true.