One of the common methods of increasing skill levels per character is via practical use in combat. For example, weapon skills can be increased by attacking or counter-attacking the corresponding weapons in combat, Reason can be increased by using offensive spells, Faith can be increased by healing allies or using white magic, etc.
Characters in Mounted classes (Fliers and Cavalry) will still gain skill exp by simply attacking or counter-attacking while mounted. All characters with assigned Battalions will also earn some exp in their Authority skill by fighting normally alongside their battalions.
For Byleth, he/she can be trained by other instructors via the Faculty Training method.
When doing Manual Instructions to your student, you can focus on specific Skills you want them to develop. When instructing them, they’ll get exp for the selected skill you want to teach. There will be factors on how much EXP they’ll get.
If they have Talent for the particular field, you’ll get a Strength bonus (blue up arrow). If you have unlocked the Saint Statues and purchased the necessary boosts, they’ll be applicable as well. If your level for a particular is higher than the student, they’ll get a Professor Expertise bonus.
The amount of EXP you’ll get will depend whether they’ll get “Good”, “Great” or “Perfect”. There will be times that they might get a “Bad” result, which will not earn them any exp and their motivation heavily drained.
There are only three Group Tasks focused on developing riding, heavy armor, and flying skills. These are skills that you can’t learn anywhere else since the classes that will require these skills are all Intermediate ones. You can assign at least two characters in this task. Characters with talent for the mentioned skills will grow faster compared to the other ones.
Seminars will spend your entire weekend and will just focus on the skills that will be taught by instructor of your choice. You also have no control over the attendees since they'll be pre-selected or fixed.