Learn Italian: Mia madre mi consiglia di comprare lo sciroppo per questo sintomo. - My mother advises me to buy the syrup for this symptom.

This sentence is an example of indirect speech. If your mother gave you a direct command, she would use the imperative form: Compra lo sciroppo! When we report that command or strong advice using a verb like consigliare or dire, we connect it to the action using di plus the infinitive verb.

Does mi act as a direct or indirect object with the verb consigliare?

It is an indirect object. In Italian, you advise something to someone (consigliare a qualcuno). So mi consiglia literally means she advises to me.

I usually see possessives with articles, like la mia macchina. Why is there no article before mia madre?

Italian drops the definite article before possessives when talking about singular, unmodified family members. If you were talking about plural family members like le mie sorelle or added an adjective like la mia cara madre, you would need the article again.

Why is the article lo used for sciroppo instead of il?

Because sciroppo is a masculine singular noun starting with an s followed by another consonant. Italian uses lo for these words to make pronunciation flow better.