One image sits under this whole family: a fence around something precious. A garden (גַּן) is ground you wall off and tend; a shield (מָגֵן) is the wall you hold up in battle; to defend is to throw that wall around a person. The same three letters give you both the gardener who cultivates and the protector who guards, because enclosing to grow and enclosing to shelter are the same gesture. This is a geminate root (the נ doubles), so the protective idea is literally built into its repeating sound. 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 two protective verbs split by register: the pa'al לָגֹן is biblical and archaic, while modern Hebrew defends with the hif'il לְהָגֵן. The pi'el לְגַנֵּן carries the other face of the root, tending a garden, though it stays rare. The passive and reflexive shells are not lexicalized for ג־נ־ן.
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.