This root grows from one of the oldest words in Hebrew: גֵּר, the stranger who joins, the resident outsider who becomes one of the people. From it the modern verb system builds the act of conversion to Judaism: to convert someone (לְגַיֵּר) and to convert oneself, to undergo it (לְהִתְגַּיֵּר). Note the term: a גֵּר here is a convert, not merely a foreigner. The same two binyanim that recruit in ג-י-ס here bring someone into the covenant. 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.
Like its near-twin ג-י-ס, this root lives in the pi'el / hitpa'el pair. לְגַיֵּר is done to another (a beit din converts the candidate); לְהִתְגַּיֵּר is done by oneself (you go through the conversion). No pa'al, hif'il, pu'al or huf'al are lexicalized: the act is intensive and personal, which is exactly what these two binyanim express.
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.