Learn German: Was hast du vor? - What do you intend to do?

It is the separable prefix of the verb vorhaben (to intend / to plan). In a main clause, the core verb haben takes the second position as hast, and the prefix vor is kicked to the very end.

How does this sentence relate to the current lesson about zu infinitives?

The verb vorhaben is one of the most common triggers for a zu infinitive clause. If you answer this question with an action, you will use one! For example: Ich habe vor, Deutsch zu lernen (I plan to learn German).

Is there a literal translation for vorhaben that makes it easier to remember?

Literally, it combines vor (before / in front of) and haben (to have). You can imagine it as having a project or idea right in front of you.

Can I just use the verb planen (to plan) instead?

You can say Was planst du?, but Was hast du vor? is the standard, natural way native speakers ask about someone's plans for the day or the weekend.