For teachers, when you go to the game, you will see all of your classes listed as well as the public world (students will not have access to the public world):
If you want to test out the quests before assigning the students, go to the public world to see all the assignments you assigned to yourself.
If you want to have your character be in the same world as your students and enjoy Tyto together, choose the class you want to be with. Click on the class you want to join and hit "Enter World". You will be able to see the characters of students in that class who are logged in. You won't get assignments but you can still play quests.