With proper punctuation the sentence should read:
Falsche Kleidung bei Regen, Schnee und Kälte, und schon tun Hals und Kopf weh.
As an alternative to the last comma you could also use a dash.
This sentence has two parts. The first one
Falsche Kleidung bei Regen, Schnee und Kälte
is a shortened construction without verb that might have a special name that I couldn't find out. Anyways, it describes some conditions or circumstances and could be understood as
Wenn man bei Regen, Schnee und Kälte falsche Kleidung trägt
or (in a way that better matches the second part)
Man hat nur einmal bei Regen, Schnee und Kälte falsche Kleidung getragen
The second part
und schon tun Hals und Kopf weh
is quite a normal phrase with wehtun as the (separable) verb and Hals und Kopf being the subject. Und serves its usual purpose as a conjunction, and schon is an adverb that refers to the first part, saying "the condition in part 1 being true, and very quickly the consequences in part 2 will follow".
So the meaning of the whole sentence is
"Wear the wrong clothes when rain or snow fall, and very quickly your head and throat will hurt."