You can export your subscriptions to another app from the Settings menu.
- Tap the Settings cog in the top left corner of any of the main tabs.
- Tap "Import / Export Subscriptions"
- Tap "Export OPML"
- A share sheet will appear and from there you can select the app you would like to export your subscriptions to. The top row will list other podcast apps that you have installed. The bottom row will have options to save your subscription list to iCloud, or Dropbox if you have it installed. If you don't see the icon for the app you expect try tapping the "More" button at the end of either row and enabling the app icon from there.