I use Antennapod to listen to podcasts, but previously had used Google Play Music and Spotify.
Finding new podcasts is usually through recommendation. Friends have recommended several podcasts, podcasters that I like have had collaborations or recommendations for other, similar podcasts, and I've picked up a couple from similar discussions as this one. Once a member of a live play podcast was promoting it where I work and I gave that a listen for a while. I don't typically use a service or discovery tool to find new podcasts, though. Sometimes I'll search a topic and see if something pops up, but usually I see one I recognize having heard about and pick that one.
Three recommendations of my own are:
Mission to Zyxx - An improvised science fiction sitcom following a team of ambassadors as they attempt to establish diplomatic relations with planets in the remote and chaotic Zyxx Quadrant… better known as the "ass end of space."
The have five seasons of their original run, and have recently started a new loosely connected series. A friend discovered this one, not sure how they found it, and we all listened to it together. Tons of fun if you have other people you can convince to listen along!
The Constant: A History of Getting Things Wrong - A podcast about exactly what it says it is: examining the bad ideas, mistakes and accidents that misshaped our world.
This was one of the podcasts I found through a similar discussion as this one. I'm a huge fan of weird and strange history, and Mark delivers. They are usually independent of other episodes, unless numbered, so feel free to hop in anywhere. I started from the beginning, and the quality has noticeably gone up, but I never had an issue with the earlier episodes.
History for Weirdos - A deep dive into the strange obscure and relentlessly entertaining portions of human history. Married couple and armchair historians, Stephanie & Andrew, discuss the often overlooked parts of humanity. Whether the subject is an obscure event that has confused historians for centuries or a historical figure that doesn't get enough credit, we have you covered.
This is one of the podcasts I discovered by searching "weird history". They have started taking longer breaks recently, while other things in life have started taking more and more time. They cover wide swaths of history from Ancient Greece to Prohibition. They are (or used to be, haven't checked recently) fairly active on social media as well, with pictures relevant to podcast episodes, or from trips they take to locations they've talked about.