There are 3 Abbey Clubs that meet during the Night portions of Midnight Suns. Each have 5 meetings, and will occur either a few days after the previous meeting or once you pass milestones in the game. You'll be contacted to arrange a meeting on the next available night (story stuff can sometimes push these back once you've agreed a meeting, particularly if you need to progress to the next chapter before the meeting occurs).
Many of the meetings also have an optional requirement. If you meet this requirement before attending the meeting, you'll receive additional ability cards as a reward.
For the dialogue within each meeting, you can pick whatever you want - it will affect friendship with the various members, but you cannot fail a Club activity.
EMO KIDS:
Meeting 1 - occurs on Night 4
Meeting 2 (Optional: give Magik Oyster Mushroom x10, Wolfsbane x10, Skill Essence x15)
Meeting 3 (Optional: give Magik Witch Hazel x10, Chicken Mushroom x10, Skill Essence x30 - gathering these reagents requires the 1st Word of Power)
Meeting 4 - does not occur until Chapter 2 (Optional: give Magik Death Cap x10, Moon Flower x10, Attack Essence x50 - gathering these reagents requires the 2nd Word of Power)
Meeting 5 - does not occur until Chapter 3 (Optional: give Magik Corpse Flower, Dryad's Saddle, Heroic Essence x50 - gathering these reagents requires the 3rd Word of Power)
Shop Class:
Meeting 1 - occurs shortly after you unlock Spider-Man
Meeting 2 (Optional: give Spider-Man Common Blueprint x1, Attack Essence x25)
Meeting 3 (Optional: give Spider-Man Common Blueprint x2, Skill Essence x25)
Meeting 4 - does not occur until Chapter 2 (Optional: give Spider-Man Common Blueprint x1, Rare Blueprint x1, Skill Essence x50)
Meeting 5 - does not occur until Chapter 3 (Optional: give Spider-Man Rare Blueprint x2, Heroic Essence x50)
Book Club - for the optional requirements you need to read a book, this is on a table next to the fireplace in the library (actually looks like an iPad):
Meeting 1 - occurs shortly after you unlock Captain America
Meeting 2 (Optional: read The Art of War)
Meeting 3 (Optional: read Kava't Koth Vol.32)
Meeting 4 - occurs shortly after you unlock Wolverine (Optional: read The Mantle of Authority)
Meeting 5 - does not occur until Chapter 3 (Optional: read The Adventures of Crash Morgan 5: The Last Queen of Deimos)
Thanks Manny C90 for pointing this out that the optional objectives are not required to progress to the next meeting.