Is there a way that only assigned individuals can see classes to register for? I want them to see the different options, but I don't want to open it up for anyone to sign up.
The best way to restrict access to classes is based on the library.
If you used an assignment profile to make the assignment, then you can simply make sure those classes are in that specific library.
If you did a direct assignment, you would still need to have the item/classes in a library for registration. Again, it will be dependent on who has access to that library. You would need to create an assignment profile that only captures the individuals assigned that item.
I second Heather L.'s comment. That's how I would configure as well.
In order to have a specific population to only view the item is to assign it to their Learning Assignments either as recommended or required (depending on the learning needs). therefore, using and assignment profile/curricula or data import to learning assignment.