Learn Dutch: Geef de boodschap aan je moeder. - Give the message to your mother.

This is a command, telling someone to do something. In Dutch, you make commands by simply using the stem of the verb (geef) and putting it at the very beginning of the sentence. You do not add a -t and you do not use a subject pronoun like jij.

I learned that boodschappen means groceries. Why does boodschap mean message here?

The singular word boodschap usually means message or errand. When you use it in the plural as boodschappen, it almost always means groceries (referring to the errands you run at the supermarket).

Why do we use aan instead of naar in this sentence?

You use aan when giving, sending, or showing something to a person. You use naar when talking about physical movement or direction toward a place, like walking to the store (naar de winkel). Since you are giving the message to someone, aan is correct.

Could I write this sentence without using aan?

Yes! Just like in English (Give your mother the message), you can drop aan if you change the word order. You simply put the person receiving the item before the item itself: Geef je moeder de boodschap.