It all really depends what device you are listening from.
The iPhones have a podcast app that allow you to subscribe, and when a new episode is available, it pops up, to listen any time.
Android phones have similar apps. The best I found was "Podcast Addict". This one does the same thing as the iPhone, but it is made specifically for android.
Also, most podcasts are available through iTunes. And if you can't find it there, and you are able to download it, you can then save it to your computer, add it to iTunes, and then put it on your phone.
This is how I enjoy my podcasts. They are what get me through the day. I can only listen to so much Pink Floyd.