If you’re just getting into podcasts and want a basic player, I’d recommend giving Overcast a try (it’s free, after all). And for those who want more bells and whistles, Downcast definitely has it dialed in.
I am surprised that Instacast is not in the list. It’s certainly older than the majority of podcast apps highlighted in this piece and for me (and I have tested all of them) has the best syncing service between iOS devices. They even have a Mac client as well.