For Android there are lots of open source apps to pick from, I find it difficult to recommend a particular one because everyone’s got different tastes. AntennaPod seems to be the most popular as far as I can tell but I’d recommend trying several and see what you like best. Best is probably to install F-Droid (kinda like the Google Play Store but for free and open source apps) and just search for “podcast”. They also warn about what they call “anti features” in apps, for example some apps use Apple APIs in the background for podcast directories/recommendations.