There are several quest lines and relationships/characters that require you to go to school. The school is quite an important part of the game, but you will not get punished if you skip it sometimes.
You can help Mr. Carter in the morning if you get up early and go there directly. Then you can still attend school. I played the game that way that I saw school as the main job and attended it whenever possible. But there are some things that require you to skip school if you want to progress there.
I hate that part that talking to Corona will force you to fail school on that day. How long can a short chat with the post box take?