I also have a droid. There are two ways to listen on a smart phone: streaming directly to the phone, or as buzzbee mentioned, downloading the podcasts to a computer then transferring the podcasts to the smartphone and playing them as you wish.
Streaming can be done with free apps like Stitcher (my fav) or others. Its pretty easy to set up, and you can add podcasts to a Favorites Station to automatically update. It plays podcasts as well as internet radio stations etc. Keep in mind that this may use up data minutes, so think about what kind of plan you have. The other way is to use something like iTunes to download to podcasts to a computer then just transfer the files to your phone via wireless or wire and play them as you wander around in your yard.
Two excellent sources for android software are Google Play and Amazon. There are a billion free and low cost apps. Hope this helps.