The problem I have with most podcasting apps is that they assume I want to always download the most recent episodes of podcasts to which I am subscribed. I would prefer to only download episodes to which I actually want to listen. After getting frustrated tweaking settings in apps like Instacast and Overcast, I've resorted to the following routine:
I then right click on the mp3 link and copy the link location to my clipboard.
When I receive the push notification (the Pushbullet app on the iPhone is of course required), I swipe on it, and the link immediately opens to...
GoodReader is an excellent app for downloading just about any media: mp3s, movie files, images, html files, PDFs. When an URL is preceded by a
g, it tells Safari to go to GoodReader, and then GoodReader downloads the file.
This app remembers where I leave off in the middle of listening to a podcast -- although I have to hit the "Stop" button after pausing the recording -- so I can easily resume the podcast just by tapping on the filename. When I'm done, I swipe to delete.