At the core of this root is the same act for the ancient ear: to sing and to compose verse were one thing, because a poem was sung. So the simple verb is "to sing", the intensive pi'el is "to write poetry", and the noun שִׁיר means both song and poem at once. A side branch of the family carries a different shade, what is left over, the remainder, since the same letters spell a separate everyday sense. 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 pa'al here is a hollow verb (middle י), so it keeps the long vowel: לָשִׁיר. The pi'el לְשׁוֹרֵר shifts the meaning from plain singing to deliberate poetic craft, which is why the poet is מְשׁוֹרֵר and not a "singer". The other 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.