I love and use:
RSS Feeds for News (each app has it's own advantages and some have offline feature):
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Heartfeed (free, 2.99 dollar/euro in-app full version)
- the look and feel are like the former Pulse News app and has a few fans in
this topic.
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Feedly (free)
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Newsify (free, in-app features to remove ads among other features, has offline feature as well) I'm not using Premium but did pay 2.99 dollar/euro once to get the ads removed.
Podcasts:
Downcast (2.99 dollar once, in-app are there but not needed it's just tips for the developers)
I have also tried others like
Overcast (free, 9.99 full version) and
PocketCasts (3.99 dollar) but always got back to Downcast because of the great settings such as a smart offline podcasts lists that get's automatically updated and only shows my offline podcasts and when I listened to one it gets removed automatically from the list/app.