I'm going to step out on a limb here and say this about that: The Hebrew writer does not mean that everyone should reach the point that he should be a teacher in a formal classroom setting. If he did mean that, then why did James write: Not many of you should become teachers, my brothers, for you know that we who teach will be judged with greater strictness. (Jas 3:1)?
What I believe the Hebrews passage means is that sooner or later we ought to have the knowledge to be able to teach. Whether we should stand up in front of a class is another question entirely.
BUT—the New Testament does teach that we should be able to do things that fall somewhere in the "teaching" area.
We should all reach the point that we can handle the problems life throws our way. For you have need of endurance, so that when you have done the will of God you may receive what is promised. For, “Yet a little while, and the coming one will come and will not delay; but my righteous one shall live by faith, and if he shrinks back, my soul has no pleasure in him.” But we are not of those who shrink back and are destroyed, but of those who have faith and preserve their souls. (Heb 10:36-39)
By learning to persevere in this way, we become good examples to others. Show yourself in all respects to be a model of good works, and in your teaching show integrity, dignity, and sound speech that cannot be condemned, so that an opponent may be put to shame, having nothing evil to say about us. (Titus 2:7-8)
We should be able to give good advice. Where there is no guidance, a people falls, but in an abundance of counselors there is safety. (Prov 11:14) Let the older teach the younger, Titus adds in chapter 2.
We should be able to correct the wayward. Brothers, if anyone is caught in any transgression, you who are spiritual should restore him in a spirit of gentleness. Keep watch on yourself, lest you too be tempted. (Gal 6:1)
We should be able to answer the unbeliever. …but in your hearts honor Christ the Lord as holy, always being prepared to make a defense to anyone who asks you for a reason for the hope that is in you; yet do it with gentleness and respect, (1Pet 3:15)
Not all of us will have the ability to organize a good lesson, the materials on hand to do the research that formal teaching may require, or the talent to keep an audience interested for long periods. But all of us are commanded to reach the point that we can teach in some capacity, whether over a backyard fence, across a coffee table, or perhaps just by being what we ought to be every minute of every day. When our "practice" has not been "constant" enough to enable us to even "discern good from evil," something is dreadfully wrong, and it's no one's fault but our own.
Therefore let us leave the elementary doctrine of Christ and go on to maturity… (Heb 6:1)