There are lots of really good science-based podcasts for kids, and I'll get to them in the not-too-distant future as I start to review podcasts for kids, but I'll start off with Shabam! because who knows how long before the zombie outbreak takes over?
I should probably explain that Shabam!, produced by a collective known as Foolyboo, differs from a lot of science-themed podcasts in that it folds its non-fiction topics -- pathogens, epidemics, cognitive biases -- in a fictional wrapper, that of a zombie outbreak. I think the mix serves the podcast well -- the fiction does provide sufficient context for the non-fiction components, while the non-fiction parts ground the fiction (which I'll remind you is about a zombie outbreak, so take the phrase "ground the fiction" with a grain of salt). It's possible to tell interesting stories about, say, epidemiology, in a non-fiction setting, of course -- yay, John Snow! -- but this fictional story works well for that subject, too.
I'm going to peg the age-appropriateness of the show at best for kids 8-13. That's based on the science and zombie-ishness thus far, which isn't for younger kids, but definitely lacks Walking Dead-like grossness. The iTunes link for the show is here. Episodes are roughly 25-30 minutes in length, and it's as yet unclear how frequently they'll be released -- I'm guessing monthly from here on out. Aside from a brief statement near the beginning and the end of the show that it's partially sponsored by Google, there are no ads during the show. (Also of note: they do a really good job of editing the closing credits, telling the story in between all the various credits, giving the listener a reason to not hit the fast-forward button.)
The other reason I wanted to lead off my podcast reviews with Shabam! is that in its narrative structure, it's going to be best if you listen from the beginning. Since they're only 3 episodes in, now is a good time to jump in and listen with your kids. You know, before the zombie outbreak hits your town.