CNET’s (June 2014) list of Android podcast apps worth a listen and a look includes:
DoggCatcher ($4.99)
Pocket Casts ($3.99)
BeyondPod (free)
Podcast Addict (free)
Stitcher Radio (free)
TuneIn Radio (free)
iOs (iPhone and iPad) apps include:
itunes (free from Apple)
CNET’s (2012) article, Three podcast apps that are better than Apple’s lists