Hey Marie,
While specific recommendations for other systems don’t quite come to mind, you might indeed be able to use The Events Calendar and a ticketing add-on to enable these features on your site.
It seems like a lot of the functionality you need is core to WordPress itself. If you make user accounts for your instructors and give them enough access so that they can publish events, and create tickets for their classes (even just free tickets, $0, just so that people have to “sign” up for them and reserve a spot and such), it might work well for your project.
You can learn about that process of creating Author users on your site in other articles online like this one, for example → http://www.wpbeginner.com/beginners-guide/how-to-add-new-users-and-authors-to-your-wordpress-blog/. Search around a bit and you’ll find a ton of great resources on that topic.
I hope this information helps! Let us know if it does, or if you have any other specific questions here.
Thanks!
George