Adam has loved the following podcasts that are designed for kids:
Circle Round - produced by WBUR public radio in Boston, these high production value podcasts feature professional actors telling folk tales from around the world...they do have advertisements, though
Reading Bug - produced by an independent bookstore in San Carlos, this features a different "reading adventure" inspired by children's books, along with songs and activities for kids. Good for a rainy day at home—have paper and crayons/markers/pencils available.
Sound Detective - fun show involving sounds, starting Levar Burton (of Star Trek & Reading Rainbow fame)
Samsh Boom Best - fun "debates" for kids like "which is better, Ice Cream or Pizza?"
Aaron's World - a DIY production by father-and-son (and their friends/family), this is a bit goofy (there are way too many sound effects) but especially the early episodes feature interesting facts about dinosaurs
Story Nory - a British company produces these oddball stories, some of which Adam really liked
Stories Podcast - we haven't tried this yet but some people like it
Tales from the Lilypad - cute fairy stories, very soothing (good for putting your kid to sleep)
Recommended with Reservations:
The Adventures of Red Knight - a bit frenetic (characters talk really fast & there are jokes almost every other second) but Adam was obsessed with this for a while
The Alien Adventures of Finn Caspian - a DIY production a bit like Aaron's World but with a sci-fi alien world; there's also a spin-off "BeeBop Tales"
...not technically podcasts but we use these in a similar way:
Dealing with Dragons (and sequels) - these books by Patricia C. Wrede are funny and subvert many tropes of fantasy, especially around gender; the first is especially good
Land of Stories series - these tongue-in-cheek fairy tales are written by Chris Colfer, who is most famous for playing Kurt on Glee; they're fun and engaging
I'm not a fan of screen time in general, but if your kid is going to be on a screen, why not make it educational? Here are some we've tried:
ABC Mouse - this is a comprehensive learning app that teaches math, reading, and other subjects; Adam basically learned to count from 10 to 100 using this app
while it's expensive, you can get it for free from many public library systems through the "Bring Learning Home" program...many public libraries in California (e.g. Sunnyvale) let you sign up as long as you're a resident in the state
DragonBox games - we've been a big fan of these math-focused games; they all cost money but aren't too expensive
Algebra - really teaches basic algebra to kids as young as 5 years old in a fun way, no kidding!! (iOS, Android)
Big Numbers - beautifully produced game that teaches long addition & subtraction, it's really fun (iOS, Android)
Elements - a geometry game that teaches about triangles & quadrilaterals; it's not bad but has a bunch of arbitrary rules that make it a little bit frustrating (iOS, Android)
Khan Academy Kids - free, well-produced app that's good for very young kids (it's supposed to be for kids 2-7 years old, but at least when we tried it in 2020 when he was 5, Adam got bored with it, iOS, Android)
Recommended with reservations:
PBS Kids - the good news is that this is free...the bad news is that some of the games are only barely educational
Originator games - this company makes pretty high-quality games, but some of them are a bit addictive and/or expensive (with "expansion packs" and/or subscription fees)
Endless Numbers - learn numbers
Endless Alphabet - learn basic word sounds
MathTango - addictive with a subscription fee, but fun/educational
Prodigy - a retro Legends of Zelda-type game that features "wizard battles" where you solve math problems; you frequently get upsold on Premium Memberships (iOS, Android)
we told Adam "This game teaches you two things: math and saying 'no' to advertisements. If you ask to get a Premium Membership, we'll uninstall the app" and that seems to be working so far...
NOT recommended
Math Blaster - much worse than the old-school game; most of the game doesn't even involve math
In general we have had much more luck with device-enforced screen time (e.g. on iPads you can go to Settings --> Screen Time to set overall or per-app limits). This way it's not the parents/caregiver nagging the child to stop, it's the device itself saying it's time to stop. Somehow this has been easier for our kid to deal with...