Learn German: Trink den Tee gegen die Erkältung! - Drink the tea against the cold!

This is a command for one person you know well (the informal du form). To make this, take the normal du form of the verb (du trinkst), drop the -st ending, and remove the word du. That leaves you with just Trink!

Why does the sentence use den Tee instead of der Tee?

Tee is a masculine noun. Since the tea is the direct object receiving the action (it is the thing being drunk), it must be in the accusative case. For masculine nouns, der changes to den.

We take medicine "for" a cold in English. Why does German use gegen (against)?

In German, you fight illnesses rather than support them! If you said you were drinking tea für die Erkältung, it would sound like you are drinking tea to help the cold grow stronger. You always use gegen (against) when treating an illness.

What case is die Erkältung in here?

It is in the accusative case. The preposition gegen always forces the noun following it into the accusative. Erkältung is a feminine noun (which you can easily guess because it ends in -ung), and the feminine article die stays die in the accusative case.