Mission Mode is the story mode portion of the game. Each story has multiple chapters to them. Stories are unlocked once you complete the previous one.
This is also a fairly new gameplay, so stories are still being added.
Basically, you walk around the school to find certain kares that you need to talk with or get something from.
Clicking the blue arrow with the pink flag icon will take you to the kare you need to speak with.
Click the kare with the pink flag icon on him to talk to him.
Clicking on the blue arrow with the orange coin purse will take you to the kare who has the item(s) you need.
You will also occasionally be taken inside the school. Click the yellow arrow to go in/out and up/down floors.
Occasionally you will see other kares while walking around. You can click them to speak to them and get their N card.
Mission Mode uses up genki, so be warned.