Learn French: Ils achètent une surprise pour l'enfant. - They are buying a surprise for the child.

In casual French, you can just keep the normal sentence order and replace the object (une surprise) with the word for 'what' (quoi). So you would say: Ils achètent quoi pour l'enfant ?

How would I ask 'Who are they buying a surprise for?' using that same trick?

You do the exact same thing! Leave the sentence as is, but replace l'enfant with the word for 'who' (qui). Keep the preposition pour right where it is: Ils achètent une surprise pour qui ?

Why does achètent have an accent mark when the infinitive verb acheter doesn't?

Acheter (to buy) changes its spelling slightly in certain forms to match how it is pronounced. For the ils form, we add a grave accent to make an 'eh' sound (ah-SHET) instead of the weak 'uh' sound from the infinitive.

How do I know if enfant is masculine or feminine here?

You actually can't tell just from looking at l'enfant! The word enfant can be masculine (un enfant) for a boy or feminine (une enfant) for a girl. Because it starts with a vowel, the words le or la both shrink to l', so the phrase looks the same either way.