When you begin Fire Emblem: Three Houses, you'll join the monastery as a member of the faculty, as one of the professors, taking on the duty of managing one of the classes of students and warriors that study there. Because the protagonist, Byleth, is a professor, they can't level up their skills based on what is taught in lessons. This means Byleth grows slightly differently, especially where things like Budding Talents / Hidden Talents are concerned.
Faculty Training is the answer to this problem - essesntially, Byleth goes and spends time with the faculty characters, the other professors of the academy, and has private faculty teaching to level up skills.
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Fire Emblem: Three Houses Faculty Training - where each skill can be learned in Faculty Teaching
First of all, it'll pay to remember that Faculty Training is an action that you can undertake in your free time exploring the monastery, and it'll cost you an action point - so you can't do this in an unlimited fashion, so keep that in mind.
In order to undertake Faculty Training, all you need to do is go and speak to the professor in question and select the faculty training option. Each professor specializes in a few different skills, which we'll detail below. If you want to know more about what each skill does, check out our Fire Emblem: Three Houses class guide for information on class mastery, skills, abilities, certification and more.
One other thing to remember is that Faculty Training can grant an affinity bonus between Byleth and the teacher, depending on who the teacher is. That can be useful if you have plans to pursue one of the teacher that is also a romanceable character in Fire Emblem: Three Houses.
Also, Three Houses Faculty Training actually splits into two categories - before and after a key event in the game, the time skip, and after the time skip. Nintendo announced and showed parts of the time-skip prior to Three Houses' release; without getting into spoiler details, at a point in the game the timeline will skip forwards a bunch, and at that point things will change - including faculty training. Here's how it breaks down:
Faculty Training before the Timeskip: what skill each teacher can train Byleth in
These are the Faculty Members than can teach Byleth to level up in skills. Remember that this costs time/action points during exploration of the monastery. In addition, get affinity up depending on who trains you.
- Jeralt - Lance, Authority, Riding
- Manuela - Sword, Faith, Flying
- Hanneman - Bow, Reason, Riding
- Seteth - Sword, Lance, Axe, Authority, Flying
- Alois - Axe, Brawl, Heavy Armor
- Rhea - Sword, Brawl, Reason, Faith
- Catherine (available from Month 7) - Sword, Brawl
- Gilbert (first available from Month 8) - Lance, Axe, Heavy Armor, Riding
- Shamir (first available from Month 7, then Month 9) - Lance, Bow
Training Byleth after the Timeskip: how things change
After the timeskip where time moves on significantly, thanks change at the monastery and for Byleth. The good news is that the equivilant of Faculty Training still exists - and it's actually more broad. The bad news is that there's a semblence of random chance to it.
After the timeskip, any character can train Byleth, but the exact skills they offer to train will depend on both that character's stats and Byleth's stats. The game will make it clear what somebody has to offer - so chat to everyone and shop around before spending time training with them - and don't forget that affinity bonus you can also sometimes get.