If you're registering into a new class but you already have a Student account, you can simply add the new class to your account. Click on the menu in the top-right corner and select "Join a Class".
Enter the new Class Code and you'll be enrolled into the new class!
Note that because you're in multiple classes now, when you first login you'll see a dropdown to let you select the correct class. Your Missions are class-specific so always check which class is loaded.