I’d like to maximize my professor level as early as possible to max out my availiable supply of activity points. What activities in the monastery offer the most professor XP per activity point I spend on them?
Tl;dr: Tournaments > Choir Practice/Sharing a Meal > Cooking. Faculty Training gives no XP. Gardening yields XP at zero activity point cost, but throttled to once per week. Fishing is effectively infinite/limited by bait, not AP.
Regardless of which of the many actives you do, increasing your professor level will be time consuming and take a good bit of grinding. However, the reddit community has a collection of all the activities and the amount of experience they provide.
Fishing provides some experience, and does not consume any activity points. So this can be a grindy way to increase your professor level, given you have the patience. The experience that is provided varies on the fish that is caught.
In addition to reddit, this guide recommends this route through breeding season. You can check whether or not it is breeding season by the fishing icon on the calendar. During this season, you have the chance to obtain multiple fish with a single bait. This will increase the amount of experience relative to the number of fish you catch.
Gardening will provide between 100xp and 300xp, depending on the cultivation you select when planting the seeds. The downside to this though is that you can only cash in the experience the next time you visit the monastery.
Regardless of motivation or support levels, sharing and cooking meals will provide 200xp and 150xp respectively. However, since this costs activity points, it's recommended to use them sparingly with units you which to build motivation and support with.
Regardless of motivation or support levels, participating in choir practice will provide 200xp.
Winning the weekly tournament will provide 300xp. Unlike other activities, this one is repeatable! So you can continually obtain 300xp for each activity point you use
Occasionally you can find items - such as books, tea leaves, or food - around the monastery. On the map, these are indicated by a blue sparkle (just like lost items). When you pick these items up, you will gain some tidbits of information and anywhere between 100xp and 500xp
Towards the beginning of each month, you are given the chance to answer questions some of your students may have. You are guaranteed at least 1000xp for answering the question. If you answer correctly and the student's motivation increased by 50 points instead of 25, you will obtain 1200xp instead.