Hi Alysia, thanks for contacting us!
The Events Calendar (and Events Calendar Pro) do not ship with an [events] shortcode … in the normal run of things the events calendar is made available at:
http://example.com/events
It is not however a real “page” in the traditional WordPress sense. Instead, The Events Calendar generates this page dynamically. So, some things to try:
1. Can you confirm which versions of The Events Calendar and Events Calendar Pro you are using (you can see the versions by visiting the plugins page)?
2. If you have a normal WordPress page, post or category named “events” there is a possibility that this is conflicting with The Events Calendar. Try trashing this item or renaming it (and renaming the slug).
3. Try deactivating all other plugins except for The Events Calendar and switching to a default theme such as Twenty Eleven (just temporarily) and check if the problem persists.
Let me know how you get on!