"Andi is reading out the new school rules."
Translation:Andi membacakan peraturan sekolah yang baru.
Without "yang," would this mean "the rules of the new school"?
Yes, that's correct.
'peraturan sekolah baru' ==> the rules of the new school.
'peraturan sekolah' ==> the rules of the school.
'yang' refers to the head noun (peraturan).
'peraturan sekolah yang baru' ==> the rules of the school, which are new.