By default, when you sign into iCloud with your Apple ID on your new device, iCloud calendars and Reminders will show up again but not 3rd party calendars such as Google or Yahoo...
If you have any 3rd party calendars, you need to get them to sync with your device manually by adding the accounts in the iOS settings area below.
iPhone/iPad settings>Mail, Contacts, Calendars.
If you add your 3rd party calendars accounts in there, your device will start syncing with and showing them on Awesome calendar.
In case of note data, it is a little bit complicated, technically you can choose where you save the note data, for instance, you can save it to iCloud or Local DB.
There is an option named Use iCloud for Notes in its settings page.
If you have turned Use iCloud for Notes on, the note data have been saved on the iCloud and you will be able to restore the data when you turn the option on again on your new device but have not turned on and you do not have the old device what you have been using, unfortunately there is no way to restore the note data because they are gone with your old device.