One root holds a small surprise: new and month are the same three letters. A month (חֹדֶשׁ) is named for the new moon, the moment the sky renews itself, so the calendar word is literally a renewal word. The verb makes things new again (innovate, renovate); the reflexive lets a thing renew itself (regenerate). Around them sit the adjective "new", the noun for an innovation, and the abstract act of renewal. Once you see that the month is just "the renewal", the whole family snaps together. 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.
This root lives in the intensive binyanim, not the simple pa'al: the everyday verb is the pi'el לְחַדֵּשׁ (to renew), with its reflexive hitpa'el for things that renew themselves. There is no common simple "to be new" verb; the adjective חָדָשׁ covers that state directly. Not every root uses the pa'al, and that is normal.
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.