One idea runs underneath: starting something on its right path for the first time. Start a child on the way and you educate them; start a building into use and you inaugurate it. These are the same act in Hebrew, which is why חִנּוּךְ (education) and חֲנֻכָּה (the festival of re-dedicating the Temple) are siblings. Even the eight-branch chanukiyah is named for that first lighting. Each word below is a first-beginning of one kind or another, 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 two senses divide by binyan: pa'al לַחֲנֹךְ inaugurates a thing (a building, a season), while pi'el לְחַנֵּךְ inaugurates a person into life, that is what education is. The nif'al is the plain passive of the inauguration. No pu'al/huf'al/hitpa'el are lexicalized here, which is normal.
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.