Podcasts are part content, part subscription, and part playlist. The've never fit neatly into iTunes, even though there were practically born in the iOS world.
Believe it or not, native app support is better than it was, but still falls short of some of the better third party apps. Of which there are many. Not ones I've used myself. Overcast is the only app I've used besides the native Podcasts app, mostly because I'd been a fairly light podcast listener before I got Overcast.
Here is a list of some of the better podcast apps; the ones I've seen recommended.I haven't doen enough research to know which of these have native iPad apps (not just iPhone versions). The ones that don't will probably work on the iPad, but not look all that great.
Overcast FM
Downcast *
iCatcher
PocketCasts
Pod Wrangler
Instacast *
Castro
* These appear to have Mac apps in the Mac App Store (for the desktop). They may have Windows apps too. No idea. If they do, there is a fair chance they will sync with the iOS version.