There’s a thread here on the forum called How do you listen to GeekNights? where people have talked about podcast apps and recommendations.
I use Overcast, and it’s great. It’s like the perfect app for me now, because all of my issues with it over the years have been solved, one by one. It just keeps getting better, so I don’t even mind paying the $10 per year subscription so it doesn’t show ads. I follow the developer account on Twitter, and plugged away at a few small issues over the years, like having a dedicated panel for podcast chapter selection, and every problem so far has been solved (even if not in the way I suggest).
The playlist stuff is good, but now that is bare minimum table stakes for a podcast app. The latest big improvement to Overcast is the search functionality. It used to be useable, but quite slow. Now it has google-style auto-suggestions. It’s so quick it feels like magic, and gets the most obvious result examples for you before you expect.