There are no specific days that any of the main stories need, and any specific times are listed clearly and multiple times in dialogue, the Leads menu and in the girls' Hint section on their Heart cards.
Likewise, the requirements for the quests are listed clearly in either the Leads menu or in the girls' Hint sections.
There's a lot of legitimate complaints that you can make about CK, but "the main plot is locked behind specific days and times that you're not told about" isn't one of them ;)