How can I download an MP3 to my phone instead of opening it in a music player?
Are you able to Chrome-to-Phone a page containing the link to the mp3 you want? If so, Dolphin (and most most every Android browser) will allow you to "save link as" (or similar) when you long-press on it. This will download the file instead of trying to play or stream it.

More cumbersome solutions include downloading the mp3 you want on your PC then transferring via Dropbox or syncing via doubleTwist (or even their AirSync for-pay feature). Doubletwist even has its own built in podcast features much like iTunes. BigCaseyDog's answer about Amazon MP3 would be a very similar alternative to Dropbox.

FWIW, I'm a Pocket Casts fan for the purposes of this very question - I was stuck with the same problem you describe until I fell in love with their download feature, among other things.