From the main tab in CastBox you can quickly check out the highlighted and trending content, or use the search box to instantly bring up any podcast. Thanks to the keyword indexing of searches, you can find podcasts (or podcast episodes) on subjects of interest in a simple way.
Of course, you can stream any track right there to listen to it, or download it to your local memory to access it offline later on. You can even sync playback of an episode on several different devices, so long as you're logged in on all of them.
CastBox is an excellent podcasts app with a colossal catalogue of programs. And best of all, you can upload your own podcasts to distribute for free. It doesn't cost a single dime to post your own episodes.
Requires Android 4.1 or higher