Learn Italian: Mio padre mi consiglia di prendere lo sciroppo. - My father advises me to take the syrup.

This sentence is reporting a piece of advice. If your father spoke directly, he would use the imperative form: "Take the syrup!" (Prendi lo sciroppo!). When converting a command or advice into indirect speech, Italian uses an introductory verb (like consigliare) + di + the infinitive verb (prendere).

Is mi a direct or indirect object pronoun in this sentence?

It is an indirect object pronoun, meaning "to me". The verb consigliare (to advise) requires an indirect object for the person receiving the advice (consigliare a qualcuno). Your father is giving advice to you.

Why doesn't Mio padre have an article in front of it, like Il mio padre?

Italian drops the definite article before possessive adjectives when you are talking about a single, unmodified family member. Because padre is a singular family member, it is just Mio padre, not Il mio padre.

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

You must use the masculine singular article lo for words that start with an "impure s" (an s followed by another consonant). Since sciroppo starts with sc-, it takes lo.