You'll start off with your own set of students, depending on the House that you chose to Support. Some story-related characters will join your party as you progress in the story. However, you can still recruit students from other houses and your fellow professors (except your father of course), provided that you have met their requirements. They’ll usually be interested in joining you once you’ve developed the specific Stats and Skills they’re looking for. There are also specific students that you can’t recruit, due to their steadfast loyalty to their leaders.
On top of meeting their requirements, you also need to have good Support level towards them. This can be done in a variety of ways. You can invite them as Mission Assistants then set them as Adjutants to your character. You can give them Gifts, give back their lost items, or include them in other bonding activities like Choir, eating at the dining hall, Tea Parties, etc to increase their Support Level. This is what you should do early in the game.
Once you get their Support Level to rank B, there's a chance that they'll ask you if they can join your class during the weekdays, even if you failed to meet their recruiting requirements.
You have limited attempts to recruit students per weekend. If he/she declined your invitation, you won’t be able to recruit him/her again on that same day. That’s why it’s recommended to save your game before attempting to recruit a student so if he/she rejects you, you can just reload the game. It’s also worth noting to have Activity Points remaining so you can level up a particular skill required by the student you want to recruit.
Also, as part of the story later on, some of the characters might leave your party due to their allegiance to a specific faction. We will update this list as we make some progress with the game.
Ingrid: Dex, Flying (D)
Felix: Spd, Sword
Sylvain: Cha, Reason
Mercedes: Mag, Bow
Annette: Mag, Faith
Ashe: Cha, Lance
Dorothea: Cha, Authority
Bernadetta: Str, Bow
Caspar: Str, Brawl
Ferdinand: Dex, Hvy Armor
Petra: Dex, Riding
Linhardt: Mag, Reason
Hilda: Charm, Axe*
Raphael: Str, Hvy Armor
Ignatz: Dex, Authority
Marianne: Mag, Riding
Lysithea: Mag, Faith
Leonie: Str, Lance
Lorenz: Charm, Reason
* Hilda cannot be recruited in the Crimson Flower route, can only be recruited on the Silver Snow route. This is a very narrow time frame. Make sure to get recruit her after choosing not to accompany Edelgard and before the time skip.