Italians have a reputation for being concerned with drafts, chills, sudden changes of temperature, etc. This translates to parents often being very protective of their kids when it comes to wearing the appropriate clothing for a given situation.

There's a little song featured on Yabla all about this struggle between parents and their children on this subject.

Che senza canottiera

That with no undershirt

Poi mi prendo il raffreddore

I will catch a cold later

Captions 17-18, Zecchino d'Oro - Metti la canottiera

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BANNER PLACEHOLDER

Note the verb used to catch or get a cold is prendere (to take). It's often used reflexively, prendersi Another verb that is often used for getting sick, is beccare as in the following example. 

Ah, buongiorno. Scusate se starnutisco,

Ah, good morning. Sorry if I'm sneezing,

ma, purtroppo, mi sono beccata l'influenza.

but, unfortunately, I've caught the flu.

L'influenza è un bruttissimo raffreddore,

The flu is a really awful cold,

anzi, un po' più di un raffreddore perché ti prende tutto il corpo

rather, a bit more than a cold because it affects your whole body,

e senti i brividi e ti senti debole, ti senti stanca.

and you feel shivers, and you feel weak, you feel tired.

Captions 1-5, Marika spiega - Il raffreddore

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Marika could have said: Mi sono presa un brutto raffreddore (I caught a bad cold).

When a cold is really bad (as described above by Marika) and you have to stay home from work or school, it's often called l'influenza, even though it might or might not technically be the flu as we understand it. 

Note also that l'influenza also means "the influence" and has a verb form influenzare (to influence).

Non credo che la Francia abbia influenzato in modo determinante la mia cucina.

I don't believe that France influenced my cooking in a decisive way.

Caption 13, L'arte della cucina - I Luoghi del Mondo

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We use the verb beccare to talk about insect bites, too. In this case it isn't reflexive. The mosquito is doing the biting.

M'ha beccato una zanzara.

A mosquito bit me.

When we don't have a full-blown cold, but suspect we're about to because we got a chill, we might say:

Ho preso freddo.

(I got a chill).

The verb is still prendere (to take, to get).

BANNER PLACEHOLDER

Prendere freddo is often the reason given for catching a cold. Things Italians watch out for to avoid this are uno spiffero or corrente (a draft), climatizzatori (air conditioners), ventilatori (fans), and especially not covering up or taking a shower after working up a sweat.