Alright, I think I have it mostly figured out.
Before the day starts, you pick out a lesson plan, a conversation topic, and if you have one, an event card.
Lessons determine which stat raises, conversation cards pump up point gain (and you can use multiple of the same type at the same time), and event cards give you various special events at the end of the day.
Each day during the first "session", you get to pick between 3 options.
They will either give you a great result, a normal result, or a bad result. Not only does this change how many points you get for that day, it also changes the "fired up" meter, which acts as a global modifier for point gain (ie the higher it is, the more points you get).
Great raises the "fired up" meter by between 10-15 points, good/normal gives +5, and bad can drop it from anywhere between 10-20 it seems.
You can tell how the day went by which stamp is used (square is bad, circle is good, flower is great. Or you can just read the text ;P)
The second part of the day lets you pick a conversation topic, which will give you more conversation cards that match the selected icon to pump into points in another lesson, as well as giving you some story behind Hikari.
The final part of the day can play out in 3 ways.
If you used an event card, you'll see the related event.
If you had a "great" lesson and didn't use an event card, there's a random chance you'll see an event.
Otherwise, the day will just end.
Finally, you need to be concerned with completing goals.
You can check your goals by selecting your smartphone at the beginning of the day, and it gives you objectives such as "gain x points on any skill in 1 session" or "have 4 great lessons in a row".
Completing a goal will give you a bundle of conversation cards, and/or sometimes an event card. Completing all the lessons will give you the final goal, which is usually an outfit.
You can also re-roll your top two goals ONCE, and only once, so I suggest you save it for if you get an unreasonable goal (i.e. "gain 170 points across 2 skills in a single session")
It also looks like each time you run through 7 days you get more lessons to do, and your lesson level carries over, although I'm not sure what the point is.
EDIT:
Whoops, double post. My bad.
Thanks for the explanation. I feel the first session part is the important part to get the points, but I don't know the correct answer so I just always guess random thing from the 3 options.