The literal sense is to shoot, to throw an arrow. But the same gesture, sending something straight toward a target, becomes, in the causative, to point, to direct, to teach: you aim the learner at the truth. That is why one root holds both the rifle and the classroom. The moreh (teacher) is the one who takes aim for you; the Torah is the aim itself, the direction given. Few roots show the metaphor so cleanly: instruction is shooting the mind toward its mark. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The pa'al keeps the literal arrow (to shoot); its nif'al is the passive (to be shot). The hif'il is where the metaphor lives: "to cause to go straight" becomes "to direct" and then "to teach", the same verb gives the word הוֹרָאָה (an instruction). The pi'el, pu'al and huf'al are not lexicalized for this root.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.