In reply to BusyLizzie:
I personally found the Apple podcast program un intuitive and it doesn't surprise me that you have trouble.
I use a free program called Overcast, check it out.
In general, the idea is that you subscribe to podcasts and new episodes are automatically downloaded for you. Keep an eye on your subscriptions as it's easy to forget about a subscription while it fills your phone with episodes you aren't listening to. It's perfectly okay to not subscribe and just download a few episodes to see if you like them.
Also in general, there will be a view of the podcast with all the available episodes listed as well as another view that shows the files that you have downloaded. This is where you would play the files you've downloaded.