One forceful idea: to drive out. The pi'el לְגָרֵשׁ is to expel or banish; turn it reflexive and you get לְהִתְגָּרֵשׁ, to divorce, literally to drive one another apart. From there the nouns follow: the act of divorce (גֵּרוּשִׁים), the man who is divorced (גָּרוּשׁ). Two outliers share the letters but sit in unexpected places: a מִגְרָשׁ is a plot or sports pitch (a cleared, marked-off ground), and a גֶּרֶשׁ is the apostrophe-like mark in Hebrew orthography. Same three letters, one act of separation. 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 pi'el / hitpa'el pair is the heart of the root: legaresh is to drive someone out, and the reflexive lehitgaresh is the mutual version, two people driving apart, i.e. divorce. There is no everyday pa'al here; the forceful, intensive meaning lives naturally in the pi'el.
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.