Overcast 4+

I really like Overcast would never switch to another podcast app, my main reason is the massive time saving that comes with the “smart speed” function that simply “shortens silences” as the UI aptly tells you. But there are many other reasons why I love this app:
The developer clearly takes a lot of care not only planning his app overall and its features, but also closely monitors usage and user feedback.
His complete rewrite of his app using Swift originally got a lot of, er, constructive criticism, mainly about the streaming function having been removed. He brought it back, even though it meant a lot of additional coding in the new system and a lot of server overhead and other complications. Streaming means that you can start playing a podcast immediately before the episode has downloaded fully. It makes a huge difference to me.
Also: With the Swift rewrite, the iPad version now is absolutely gorgous and I can see myself starting to use it regularly when listening at home.
Overcast is the app I recomend to those who listen to podcasts a lot. Casual podcast users might as well stick to Apple’s basic app, but you will quickly find it indispensable if you are a regular podcast listener.

I'm sorry to say my hopes are dashed. The Apple Watch addition just doesn't work. I've tried auto sync and manual but it just won't consistently send my podcasts to my watch to play without my phone. I'm so disappointed. This is still easily a 5 star app but the Watch feature is no good at the moment. I'll keep hoping. App is brilliant- watch is not.

5.0 Update- YES! Apple Watch support!! No need for my phone to listen now, using my watch as a standalone podcast player will be amazing- I say 'will be' because I'm having issues actually transferring the podcast to my watch. With the 5.0.1 update I have managed to transfer about 5 shows but it's a little inconsistent still and I can't leave it overnight to sync. Having said this, the app is getting better and better over the 2-3 years I have been using it so I'm impressed with the improvements that are ongoing.

----I have used many, MANY different podcast apps over the years- then I found Overcast. Simple and powerful with a free (unintrusive ad supported) or paid version which is well priced but doesn't offer much benefit. Don't bother searching for other podcast apps. Just use this one- trust me!