I listen mostly to political news and media criticism shows, but I like:
Radiolab - science and philosophy reporting.
On the Media - Media criticism of current events.
Pod Save America - the relaunch of Keeping it 1600 - lefty political news from some ex-Obama staffers.
Intelligence Squared - a genuine debate show with two teams that debate national issues before an audience that then votes for a winner.
Bombshell - a few State Department/diplomatic corps economist women discuss national issues while getting increasingly drunk on bourbon.
Slate's Political Gabfest - It's slate, but it's got John Dickerson, who I trust.
Whistlestop - one of Dickerson's other podcasts, wherein he puts modern news into an American historical context.
The Weeds - seriously grognard in-depth technical conversations about national policies and law.
No Agenda - More of a fringe right news/media criticism show. Very zoo radio, but impressively produced on the fly.
Uhh Yeah Dude - great comedy podcast from a couple of dudes that basically just scour the internet and USA Today and riff on it. They're doing a live show in Seattle in a couple weeks. I think both Kevin and Joey are going. Wish I could. If you want a show that can roll from a touching tribute to the death of one's grandmother and the regret about not being a better grandson directly into "so what do you know about neanderthal pornography?" this is a show for you.