For Android devices, I suggest trying AntennaPod. It is easy to find and subscribe to podcasts in AntennaPod, and it has good helpful listening options. On Apple devices, Overcast is widely acclaimed as the best podcast aggregator app. You can subscribe to either the NLT or GNT podcasts via those apps. If you are wanting to listen to the NLT series, you might like our dedicated Daily Bible Reading Android app best.
Bulk MP3 downloading:
If you would like to download the podcast MP3 files, you can do that via apps like AntennaPod. Set the app to download 400 episodes, and you should be able to get a whole years worth if your device has the storage space. Then you will need to figure out where your app saves the downloads. For Android devices, the folder may be in the folder for your app within Android/data.