I'll start with the standards, so we can get them out of the way. RadioLab. Best storytelling, hands down. This American Life. Yes, it has gotten to the point where every goddamn story rips your heart out and makes you want to kill yourself. But still... amazing stuff. ReplyAll. Stuff geeks would find interesting, told well, by a couple of real guys. Consistently Surprising. Stuff You Missed In History Class. Two actual historians - not actors or performers - research interesting shit from history, and give you a better, wider, view of the situation than Wikipedia.
And when you're all out of Reply All, there's a whole load of TL;DR to listen to! I'll second Radiolab and This American Life for making me cry all the time. After the small town mayor episode, someone recommended to me How to Be A Girl. Do you like crying? You will cry.