Learn Spanish (Spain): Espero a la señora aquí.

The Spanish verb esperar means "to wait for" all by itself. You don't need to add a direct translation of "for" (like por or para) after it.

What is the a doing right before la señora?

This is known as the "personal a". In Spanish, when the object of a sentence (the one receiving the action) is a specific person or a pet, you must put the word a right in front of them. Since la señora is a person, we need the a.

Why doesn't the sentence start with yo (I)?

In Spanish, the verb ending usually tells you who is doing the action. The -o ending in espero means it can only be "I wait". Because of this, the subject pronoun yo is usually dropped unless you want to emphasize it.

Based on the current lesson, how would I ask "Where do you wait for the lady?"

You would use the question word dónde (where) and change the verb to the informal "you" form. The question becomes ¿Dónde esperas a la señora?

If I want to ask "Who do you wait for?", how do I do that?