I regularly listen to:
Movie BS with Bayer and Snider - There are shittons of movie podcasts online. Can't really say precisely why this ended up being the one I listened to regularly, but I do enjoy it quite a bit. The hosts have good banter, and are able to accurately convey their feelings on current movie releases without spoiling them. Also I've sent them 2 opening theme songs and they use both of them (alternating with like 4 or 5 others) so it's the only podcast I listen to that features me in a small way.
Harmontown - Dan Harmon's (of Community and Rick And Morty fame, among other things) podcast where he and Jeff Davis (and guests) ramble about nothing in front of an audience every week. Sometimes it's absolutely hilarious, sometimes it's poignant and interesting, and sometimes it completely sucks and goes nowhere. It's not going to be for everyone but it's got a unique vibe to it. It's not too hard to jump in to the podcast as it is currently. But I'd highly recommend going to the start and listening to it in order since the show gradually develops it's own various narrative through-lines and running jokes in regards to their personal lives, the guests that appear, and the structure of the show as it progresses. Also in the early days they start a D&D campaign as a segment of the show and hearing that progress is one of the best parts of the show.
Hello Internet and Cortex - Both of these podcasts feature youtuber CGP Grey, the first is him and youtuber Brady (from Numberphile and Periodic Videos) having some really enjoyable two-guys-talking banter about various topics. The second is him and a podcaster named Myke Hurley and that one is more focused specifically on work productivity and technology. I definitely prefer HI since the topics are more broad and Brady is the more entertaining guy of CGP Grey's co-hosts, but there's naturally a lot of crossover between the two. Both are very accessible to a wide audience.
The Pro Crastinators Podcast - my favourite but also BY FAR the least accessible. The Pro Crastinators are a collective of amateur and semi-obscure youtubers/internet artists of various types (reviews, games, anime, comics, vlogs, satire, comedy, "nerds shooting the shit" stuff). Each one the 11-members can talk your ass off about anything, and all of them have uniquely strong, intense, and often caustic personalities. Piling a bunch of them into the same space always results in total madness.
In isolation their podcast is entertaining and bizarre, but it's really hard for me to imagine listening to it in that way since I've followed most of their output for years before the podcast started. All of them have an enormous amount of history, in-jokes, lore, alter-egos, invented characters, and other stuff going on. Having knowledge of all that enhances the enjoyment of their shows like a hundred-fold cus they don't take any time to let the listener catch up. There's so much going on that even I don't pick up on all of it and I've been following their output for years. No idea if anyone other than myself would be even able to get into this because of all the context required. But the collective taken as a whole is one of my favourite internet-things ever, and the podcast is the culmination of their collaborative output.
Switched On Pop looks pretty interesting to me. as does Between the Liner Notes. I'll have to look into both of those. I like hearing any kind of music broken down in depth like that. "Song Exploder" is another podcast kinda like that you might enjoy if you don't already.