Pocket Casts [$5.99] is a very automated podcast app with lots of features, like sleep timer and auto download.
Pro Excellent, easy to use discovery system
Clicking the plus button in the top-right corner will bring you to featured podcasts. From there you can access multiple ways of finding new things to listen to such as: top charts (that can be separated in audio and video), search (that allows searching both by name and feed URL), a list of podcast networks and different podcast categories.
They also have a great selection of podcasts to choose from. The team behind Pocket Casts work hard to maintain a reliable and high quality catalog so you can avoid out of date listings, problems with raw feeds, etc.
Pro Automated playlist creation
Pocket Casts has a feature called Episode Filter, that allows automatically creating playlists based on setting such as playing states (unplayed, playing, played), status (downloaded, downloading, not downloaded), episode type (audio, video) and of course podcasts themselves.
You can also customize the appearance of the playlist with a color and an icon to help it stand out easier.
Pro Remembers playback speed & starting position
You can set a 'start from' position for a podcast to skip a set number of seconds from the beginning of an episode. Great for when you don't want to listen to an intro you've heard a couple of dozen times before.
Additionally, you can also change the playback speed, to slow down fast talkers or speed up slow talkers.