The conditional is the future of the past. It means that, relative to the current past time frame, the action will occur at some point in the future.
The important thing is that it does not shift the time frame forward to that time. Thus, if your sentences were written with the past tense as one of the commenters suggested, then the next action would come after Miguel and his brothers going to school and his parents going to work.
But more likely than not, the next sentence is something like, Miguel se duchó y se vistió con su nueva camiseta. Notice then the order of action:
Wake up → breakfast → shower → dress → go to school / work
Nonetheless, the narrative order is
Wake up → breakfast → [will] go to school / work → [but first] shower → dress
Another commentor mentions they might use the future. That would also work: the factor involved is (1) in the present time frame, have they already gone? If so, must use conditional. And (2) if in the present time frame, are they still eventually going to go? If so, use future, but if something got cancelled and we know that they will definitely not, use conditional.
Also, whenever conditional is used as a future-of-the-past, you can replace it with ir a [verb] and it should make much clearer sense for non-natives:
Hoy Miguel se despertó temprano y comío su desayuno con sus padres y sus hermanos. Mientras sus hermanos tambien se iban a ir a estudiar y sus padres se iban a ir a trabajar, hoy Miguel iba a ir a su primer día de clases.
The double use of ir is probably why the author opted for iría. If you use other base verbs it sounds a bit better.