PowerPress by Blubrry is the #1 podcasting plugin for WordPress. Use this built-in integration to create a podcast, and it'll manage your media files, episodes, and iTunes feeds. Also use it to grant access to private podcasts based on memberships.
PowerPress is iTunes and Google Play compliant and has more than 100 features, including SEO tools and MP3 tagging. It's also consistently upgraded and maintained to retain its status as the world's most powerful podcasting plugin.