You likely can do this in the podcast apps you already use.
How Do I Download a Podcast to an Android gear?
The best way to download podcasts on Android is by using Google Podcasts, althoughAndroid has several podcasts apps.

Well stick with Google Podcasts, but these steps will broadly apply in most other apps.
In Google Podcasts, search for podcasts in the Explore tab.
It will be at the bottom in the center, with a magnifying glass icon.

The search finds individual episodes.
Next to each episode youll see an arrow enclosed in a circle.
Once you tap the download button, the podcast will begin downloading.

Tapping that will expose the download button and other options.
The download button will turn green when the podcast is finished downloading.
How Do I Download a Podcast to an iPhone?

For these instructions, well stick with iOSs default app as its already installed on your phone.
In the Podcasts app, search for a podcast using the search function.
Download Episode will be the first option in the menu.

When youre finished with the episode, press the three dots again and select Remove Download.
Can You Download and Save Podcasts?
When you listen to a podcast on your mobile unit, the files typically stay there after playing them.

The same advice applies to desktop operating systems like Windows and macOS.
macOS has a dedicated app called Podcasts and works similarly to its mobile counterpart.
That file would be stored on your gear until you delete it.
Some podcasts may be exclusive to a platform and are only downloadable or listenable through that platforms app.
You may also need a subscription to download the podcast.
The vast majority of podcasts are free for you to listen to without a subscription.
FAQ
Once you find a podcast on Spotify, you might download episodes from its page.
Click or tap theDownloadicon (a downward-pointing arrow inside of a circle) to save the episode.
Spotify stores items you download in the app.
Because NPR podcasts are available on all major platforms, you might download them from there.
Alternatively, you’re free to get them directly fromNPR’s website.