One idea sits under these words: a mark that points to something else. An אוֹת is a sign in the broadest sense, the same word the Bible uses for a heavenly omen, the badge on a soldier's chest, and the letter aleph on a page. The verb side is thin but telling: to signal (make a sign) and to consent (find a thing fitting, proper). It is a small family, but it reaches from grammar to prophecy. 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.
This root almost never works as a plain pa'al verb in modern Hebrew. The action lives in the pi'el (to make a signal) and the nif'al keeps an old reflexive sense, to find oneself in agreement, to consent. The causative hif'il, passive pu'al and huf'al are simply not lexicalized here.
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.