Maintain a submission portal with an analytics dashboard that you can access here. Just be sure to mark Megaphone as your hosting provider upon submission.
To submit your podcast feed to Apple all you need is an Apple ID and password. You can then submit your Megaphone RSS feed to Apple here.
There are a handful of prominent podcast platforms that don't pull from the iTunes Directory and require that you submit your RSS feed separately.
Google Podcasts app was released in June 2018 and includes many podcasts already. You can use this article for information on how to check if your podcast is included and, if not, how to add your RSS feed to the Google directory.
iHeart Radio creates stations for your show that are accessible online or on their app. iHeart accepts podcast submissions here
Amazon Music & Audible have added podcasts and now allow you to submit for inclusion on these platforms. You can submit your podcast(s) here.
TuneIn powers the Alexa skill for podcasts and is becoming increasingly important. Submit your show here.