One root for the act of bringing together: gathering the harvest, collecting stamps, calling a meeting. The plain verb collects things; its reflexive partner gathers people (they assemble themselves). From there grow the collection you build, the assembly you attend, and the abstract act of collecting itself. Three letters, and the whole vocabulary of gathering opens up. 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.
A clean division of labour: pa'al לֶאֱסֹף collects objects (you gather them), while the reflexive hitpa'el לְהִתְאַסֵּף is people gathering themselves, a crowd assembling. The intensive and causative binyanim are not lexicalized for this root.
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.