Learn Finnish: Oppipoika lähti hakemaan hiiltä. - The apprentice went to fetch coal.

This is the core rule for verbs of leaving like lähteä. When you leave in order to do something, the action you are going to perform must be in the third infinitive's illative case (the -maan / -mään form). Using a basic infinitive (like lähti hakea) is grammatically incorrect in Finnish.

The English translation says "went to fetch". Doesn't lähteä mean "to leave"?

Yes, lähteä literally means "to leave" or "depart". However, when combined with a -maan verb, it is the most common and natural way to say someone "went to do" something. Finnish frames this action around leaving the current location, whereas English frames it around going toward the destination.

Why is the object hiiltä in the partitive case?

Hiili (coal) is functioning here as an uncountable mass noun. Because the apprentice is fetching an undefined amount of coal—"some coal", rather than a specific measured quantity or a single countable item—the object must take the partitive case.

What are the root words inside the compound noun oppipoika?

It is made up of oppi (learning, doctrine—a noun derived from the verb oppia, to learn) and poika (boy). A "learning boy" is the traditional, literal term for an apprentice.