Calendar problems with iOS 7 right after updating the device
Make sure you have a Calendar set up in iOS, either an iCloud calendar (Settings -> iCloud) or another one (Settings -> Mail, Contacts, Calendars). If your only calendars are read-only, like Facebook Events and Birthdays, you won't be able to create events.

In iOS 7 and 8, you can check the calendars that have been set up by tapping "Calendars" at the bottom of the calendar view. If you don't see an iCloud calendar or another one you can edit, you'll need to create one.