Questions & Answers about Kauf die Medizin für mich.
Why is the verb just kauf and not kaufst?
This is an informal command (imperative) for someone you address as du. To make this form, you take the regular du form of the verb (kaufst) and drop the -st ending.
Where is the word for "you" in this sentence?
Just like in English commands ("Buy the medicine!"), informal commands in German drop the subject pronoun. The du is removed completely, so the sentence starts directly with the verb.
Why is it für mich instead of für mir?
The preposition für (for) is a "strict" preposition that always requires the accusative (direct object) case. mich is the accusative form of ich (I), while mir is the dative form.
Is die Medizin in the accusative case here?
Yes! It is the direct object of the sentence (the thing being bought). Because Medizin is a feminine noun, its accusative article is die—which conveniently looks exactly the same as its normal dictionary form.