Normally we use Google’s backup function. So if enabled in Device settings > Privacy and after installing Sleep as Android on the new device you will get all your settings, alarms and other data. This assumes you install with a fast network connection enabled.
You an also do a manual export/import process. In the Graphs tab, please use [MENU] > Backup > Export and make sure the dialog says successful. Than copy the directory/sdcard/sleep-data to you new phone’s SD card. Install Sleep as Android on the new phone and do [MENU] > Backup > Import. This should preserve your sleep data. You will need to adjust the settings manually.
Third option is to use some of the Cloud Backup methods from the SleepCloud Add-on.
Please see more on this in Backup.