|I actually jumped right into this chapter without reading the prologue (oops) but I think it stood alone on its own pretty well.
I like the world you're trying to create here, one where you can just switch on the t.v. and hear about mages being killed, as if it's commonplace. That caught me pleasantly off guard.
I'm a sucker for romance in stories, so naturally I love what you've got going here with Elyk and Lilun, especially since this is the first chapter and we can only expect their relationship to grow and have ups and downs.
You do a nice job of representing the age group of your protagonists. You perfectly captured "note speech" or a sort of SMS language feel that kids use when they were passing notes; so that accurate representation is a plus.
The story is quite interesting right off the bat, the black mage issue and her using the magic all of a sudden. I actually WANT to know where it goes.
When writing in first person, one must be careful not to overuse "I". Since we know it's first person, try your best not to start too many sentences with "I". Just describe the action. As opposed to something like, "I poured the bowl of cereal. I hated this brand but ate it anyway. I would've preferred Coco Puffs", what's better might be, "I poured the bowl of cereal, the brand leaving much to be desired, but it went down the hatch anyway. Coco Puffs would've been a nicer way to start the day though."
You rated this 13+ but I don't know if some of your profanity matches with that. Be careful. I know you're trying to be realistic though. Kids curse all the time, but still watch out.
The problems with your comma usage are lessons learned with time, more reading and study, so I won't go into them in detail here. But definitely when you edit this at a later time, better apply some of the general comma rules.
Remember that the thoughts and speech of an individual take their own paragraph, otherwise readers get confused as to who is speaking. For example, you blend Elyk's thoughts with Lilun's first person narrative here in the same paragraph. It can get very confusing to a reader.
[ "Don't Elyk, just don't." I cut him off as I wrapped my arms around myself as if I was cold. "Please, not right now." Elyk looked at me, sadness in his eyes. He could sense the crushing sadness that resided within me, as could Kairu. 'I need to talk to her' He thought. 'But I don't know how.' Elyk let out a gentle sigh and Kairu whimpered. All they could do was hope that I felt better as the day went on.]
Note: Some people also call this "head hopping" where you jump from one character's head into another. While there's nothing wrong with giving multiple characters their own POV perspective in a story, head hopping is more of a frantic, unclear leap all over the place. Transitioning from one head to another, if you choose to do so, should blend in a way that doesn't jar the reader but still let's them know whose headspace they are in.
With more reading, you'll be exposed to different ways to say the same thing. Be careful of repetition. For example, you start the chapter off by saying she "ran" and "ran faster" a lot. Change it up a bit. If you've already said "I ran faster" and want to indicate she "ran faster" again, you could use something like, "her legs fiercely picked up their pace", etc.
^_^ But your story is off to a good start, so kudos and good luck.