I'm not aware that you cannot use an iCloud Backup as described in Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support but I'm not 100% sure.
At a minimum you will be able to transfer Contacts, Notes, and other data by going to Settings > [username] > iCloud and turning on the options for those things that you want synced. That will allow the data to be synced to iCloud. When you turn on the same options in the new phone the data will sync to it. If you leave those settings turned on, any future changes will automatically sync to iCloud.