Here is a list of potential downtime activities.
Remember, if you have a close friend (level 4), they can help you (act as if trained in the skill). This is essentially the Follow the Expert skill. Note: This can only be used if you are ok with that NPC learning the information you learn from the activity you're asking for help on. Essentially, spying on another student is a difficult ask and only a few students will participate in this (such as Muruwa, Kalaggi, or Ignaci).
If you have a friend that grants a bonus to your classroom study, tell me when you send your list.
If you want to use assurance, tell me when you send your list.
If you want to do a specific task with another classmate, tell me when you send the list.
Universal
Study for your branch (Skill or lore associated with your branch)
Cram for your branch (gain 1 additional downtime activity. The GM will roll a flat d20 to check if you suffer a -1 to the next downtime activity, DC 8).
Work a job for earned income (based on job)
Spend time with an NPC (romance unlocks at level 6, first to 10 gains a unique bonus. Cannot gain 10 until passing a certain quest).
Purchasing items (no skill check)
Personal research and quests (depends on activity)
Upgrading your mask
Crafting your specialty staff or wand (required to progress your branch after Book 2)
Crafting with NPC (depends on NPC an options, typically crafting)
Sharing knowledge of rituals (based on skill used for ritual)
Club activity (based on club)
Book 2
Study Griffons
Deciphering Stone Ghost's notes
Counseling with the Anadi
Stories with Janatimo
Try to free Binji from his slave mentality. Give him a warm bath, some Das Kapital, anti-lice shampoo, anti-curse talisman, talk with someone who specializes in curses, etc.