Like a lot of people, I didn't listen to a ton of podcasts until 2010 or so. I listened to Bill Simmons' B.S. Report, and a few other ESPN and NPR podcasts. That was about it.
Then two things happened. One, I read a post by Ezra Klein, suggesting that most podcasts can be listened to at 2X speed on your iPhone. So, now a 2.5-hour rambling Joe Rogen Experience can be digested in two or three 30-minute sittings at the most in between meetings and conference calls.
Secondly, I started commuting via public transportation (30-45 minutes each way) and eventually working from home. As such, my appetite for podcasts went from "Eh, I go to the gym twice a week," to "I have several hours of silence per day which I can fill with whatever I want (preferably for free.)"
I accidentally deleted all of my old podcasts from iTunes in March, so I had an opportunity to start from scratch in terms of what I listen to day to day.
Here are the programs that I consider to be 'Must See TV' for podcasts:
1) This American Life - This American Life is one of those NPR programs that is now hard for me to imagine working WITHOUT podcasting. I suppose people used to schedule their weekends around listening to This American Life or would tape it from the radio? Or maybe it was like Car Talk, which seems to always be serendipitously on the radio every weekend when you do errands or go on a road trip. TAL has attained a certain level of notoriety for the Mike Daisey episode about Foxconn which it retracted, but it is the '60 Minutes' of podcasts.
2) B.S. Report - As mentioned, Bill Simmons and ESPN are the reason that most dudes between the ages of 25-45 even know what podcasts are. My only warning/complaint is that the schedule and topics are extremely random, so this could be disconcerting for some. A recent guest coup: President Obama. I also enjoy the Grantland podcasts, where Mr. Simmons acts as editor.
3) WTF With Marc Maron - The WTF podcast is hard to describe, but is essentially the marquis 1:1 interview show for comedians and actors, but also touches on issues of addiction, depression and mental health. A sampling of guest include Michael Cera, David Cross and Anthony Bourdain. Older classics are available as 'premium' episodes such as Louis CK, Dane Cook and Carlos Mencia. Language is often NSFW, but generally not over the top.
4) RadioLab - I am a recent convert to the cult of RadioLab, but I am now thorougly and utterly hooked. Topics generally look at the intersection of science, society and history. The "Escape" episode was pretty great. Slickly produced, the biggest complaint would likely be that they don't produce more episodes.
5) Planet Money - I believe I've been listening to Planet Money since the beginning in 2008. They have had excellent coverage of Haiti's earthquake recovery, interviews with authors like Nassim Taleb and Simon Johnson, and helped expose a dollar coin program that was costing the federal government millions of dollars. I've given a few 'business' podcasts a chance, and this is the one I've stuck with most consistently and most passionately.
The other good thing about these podcasts in particular is that they aren't especially timely, so you can load up a few and listen to them on a flight or long car trip.
iTunes has also has lots of Top 10 lists by country and subject matter which makes it fairly easy to find more podcasts that might interest you.
Feel free to leave other podcasts that you like in the comments.