How can I modify what public (users without password) can see?

By default the public can see only timetable of individual classes and summary timetable of all classes.
Only teachers with passwords can see the teacher's timetables and students and their parents can see student's timetables.

However this can by fully modified by you to your school needs in Timetable online administration after you click on "User rights".

There you can specify, what users without password can see by selecting timetables you wish to be visible for public access.
If you do not want to show any timetable for public, leave all fields unchecked.