Well, with the caveat that the selection is somewhat arbitrary:
Comedy:
Comedy Bang Bang: Character-based improv show which regularly has amazing guests. Probably the single most indispensable podcast in my feed. Lots of in jokes, though, so maybe start with the end of year best-ofs.
Spontaneanation: Improv show hosted by the incomparable Paul F Tompkins, with the sketch riffing off a discussion with that week’s guest.
Games:
Crate and Crowbar: Sort of successor to the UK PC Gamer podcast, featuring alumni such as Tom Francis and Graham Smith, and with fine frequent guests such as Pip Warr.
Idle Thumbs: Games and robot news talk show hosted by the people behind Firewatch (though the show precedes the studio’s formation). They talk about game design in a really thoughtful way.
Science:
Astronomy Cast: Does what it says on the tin. Everything you ever wanted to know about space.
Radiolab: Somewhat overproduced but always fascinating show about fascinating phenomena at the intersection of science and culture.
History/General knowledge:
99% Invisible: Ostensibly an architecture and design podcast, but really bite sized modern history lessons
Hardcore History: Pretty much the opposite. Deep dives on big subjects.
Movies:
QT3: Well, you know what this is and why it’s great.
The Flop House: The best of the bad movie podcasts, at least if you’re not going to watch the films. The hosts are extremely funny and have great chemistry, so you get some great extended riffs.
Storytelling:
The Moth: True stories told without notes is how it’s pitched. Usually some combination of funny and heartbreaking, but always gripping.
Welcome To Nightvale: Probably the most famous/popular of the fiction podcasts. Praire Home Companion meets Lovecraft, basically. Possibly running out of steam after 90 episodes, but they’ve got a bunch of spin-offs in the works (not to mention things like Limehouse and The Message which took inspiration from them and Serial).