If you want to backup the full app data (subscriptions, episode data, settings) go into Settings/Backup & Restore and press the Full Backup entry menu. Just like before the resulting backup file will be in a subfolder called backup within your storage location. You can press the share button to send this file to your Drive, Dropbox, email account.
In order to retrieve your podcast/episode thumbnails and downloaded episode you will have to copy/restore the content of you storage location into the new device.
Then finally, in order to restore after a factory reset or a device switch you just have to download the backup file to the device, then go into the app Settings/Backup & Restore, press the Restore menu entry and select the file to be restored.