I recall a similar show, with a group of kids travelling around in (I think) a futuristic transport of some sort, although it wasn't so much of an apocalyptic wasteland. I think there was a talking computer, and since it was on PBS (an American non-profit educational channel), I think some of the storylines revolved around spelling, reading, math problems, etc., although not necessarily blatant classroom stuff.
Zoom.
"Without mercy, a man is not a human being." Sansho the Bailiff, 1954
Hmm, I don't THINK that's it... Unless it was a small segment on the main show. The one I'm thinking of was a lot closer to Ark II, it was an sci-fi adventure but with a younger group than Ark II. I don't recall them breaking the 4th wall, or having the singing and audience write-ins, etc.
It seems like the cast had futuristic uniforms, maybe red & black?
It could have involved time travel, as well. Or if not, it may have been a post-apocalyptic show, and they would find ancient (or alien?) technology that they would puzzle out using educational skills.
Sounds like maybe a short lived Skatebirds series by the name Mystery Island.
Could also be Jason of Star Command or Space Acadamy all from around the same era.
There was also a live action show called Wonderbug.
I found what I was looking for (4 years later) - Tomes & Talismans (1985) - http://www.imdb.com/title/tt0467704/reference
It's on Youtube: http://www.youtube.com/watch?v=Z5Pb0BdT8Qo&list=PL109664CDBD2DB72D
It was about library science (Dewey Decimal System, card catalogs, etc.,) rather than literacy, spelling, and so on; the bad guys were called Wipers; and the kid's uniforms were orange and dark maroon, not red and black, although some shots looked reddish and black.