One option for hosting your podcast, that also handles syndication, is Libsyn.  You will need to budget for this as it is not a free service.  However, for as long as you maintain your subscription, you can host and syndicate a podcast without having to accept advertising bundled with your podcast.

Uploading and Publishing a Podcast

Click the green New button in the upper right of the dashboard.

If you are uploading a file recorded in other software, click Upload Audio

Here, you obviously upload your audio file.  But play close attention to the information you enter in the form on the New Episode page, because that determines if and how listeners encounter your podcast episode in their podcast app or on the web.

Obviously, you need a title.  Libsyn asks for a Episode Title and Apple Podcasts title.  These can be the same, and need not contain episode or season numbers, or the parent podcast name.  Just develop a short but descriptive title for the particular episode.

Under Description, add at least a few sentences describing what is in the podcast episode.  Also common are links to resources or other things discussed in the episode, what some podcasts have as "show notes."  

You need not include an author, or enter Season Number or Episode Number, especially if the nature of your podcast isn't sequential (listeners need not listen to earlier episodes to understand or benefit from later episodes).   

You can check the tooltips, and add categories or tags if you wish.  

With a Permalink, LibSyn will provide you a URL to post in various places on the web.  This can point to somewhere off of Libsyn's site, or can point to a page they provide with players for each episode, or what LibSyn calls its "blog."  We recommend choosing the Libsyn blog, since you can also always have players and links to individual episodes elsewhere;  it just adds to the places where listeners can find your podcast on the web.  

Episode Artwork is optional, but not necessary.