This root is built on the noun גַּיִס, an army corps, a body of troops gathered for a purpose. From that image of mustering comes the whole family: to recruit others (לְגַיֵּס) and to enlist yourself (לְהִתְגַּיֵּס). In modern Israel the active sense has widened far beyond the army: you גַּיֵּס money, volunteers, a whole team. The root lives almost entirely in two binyanim plus their shared action-noun. 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 whole story: לְגַיֵּס is doing it to others (you recruit them), לְהִתְגַּיֵּס is doing it to yourself (you sign up). There is no everyday pa'al here, the root works through the intensive pi'el. The matching passive (being recruited) leans on the hitpa'el reflexive rather than a separate pu'al.
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.