
I listen to a lot of podcasts, and it always surprises me that, as a form of media, podcasts are not more popular. In fact, when I have my headphones in and someone asks what I'm listening to, the usual response, after I tell them I'm listening to a podcast, is almost always,"What's a podcast?"
Podcasts began as audio only talk shows, and have since evolved into a very wide variety of genres, as well as HD video formats. There is something for everyone, more programming than many television networks, and yet podcasts remain under the radar for those not tech savvy.
So here it is, my attempt to get you guys into podcasts. First off, yes, Apple has their own app for podcasts, but it's not nearly the best out there. Downcast is the app you want:

Sure, Downcast is a few dollars on the App Store, as opposed to the free Apple Podcasts app, but it's ease of use and stability far outweighs the cost.
Here is a quick tutorial to get you started:
Once you have Downcast downloaded and installed (search for it in the App Store), the first thing to do is get some podcasts! Nearly all podcast subscriptions are free. It's actually very rare to find one that is not, so feel free to grab as many as you want.
In the upper right-hand corner, you'll find these four icons. What we need is the large addition symbol.

After hitting that, you'll see this window pop up:

If you already know the name of the podcast you want, you can search for it, otherwise select a category to browse many popular selections.
Once in a category, just select a podcast. I'll use The Nerdist as an example:

After selecting a podcast, you'll see a brief description. If you like what you see, go ahead and hit the Subscribe button. Podcasts are generally episodic, the most popular format being a weekly podcast. When you subscribe, Downcast will automatically download the latest episode. When Downcast is opened again, it will check for new episodes and again download anything new. Keep in mind, this is totally free.

After subscribing, the latest episode will start to download. The podcast will appear in your podcast list, and a new download badge will appear over the Downloads icon.

You can view the download's progress by hitting the Downloads icon in the upper right.

Once the download is finished, just select it to start playing. Congratulations! You are listening to a podcast.

Now, after you are done with that episode, you can delete it manually or have Downcast automatically delete podcasts marked as played (easily found under the settings menu). You can also browse previous episodes for download, or go search for some new favorites.

See? It's all very simple. Now go find some great podcasts!
~Ben
PS - I'll post some of my favorites soon.