Character quests are found inside your base. There will be a question mark over the different character heads while inside the base. Go to them, talk, and accept their quest. If the question mark is above the character you are controlling, then switch to a different character, and go back to talk to the character you were playing as before.
Each character has a 3 quest assigned to them, making that a total of 12 character quests.
Rachels quest tries to prove to Jack that they are in a video game.
Jacks quest tries to prove to Rachel that they are in a video game.
Luxos quest ask you to help him learn English.
Brians quest ask you to get different types of brains.