Events and Reminders are stored in iCloud or google calendar. So these data will be transferred to new iPhone automatically, if you set up same iCloud account.

If you turned on "iCloud for Note" in AwesomeCalendar Settings, diaries will be stored iCloud. You don't need to transfer data manually. 
But if you didn't turned on the option, these are stored in local database. You need to backup the local database and restore it. About backup the local database, refer to below link.

Unfortunately, all stickers are stored in local database. This means you need to back up and restore local database manually to transfer stickers from old iPhone to new iPhone.