The Big Bang Theory Theme.
I love The Monarchs song. Informative and fun to sing along with.
At one point in school I was required to memorize and recite the Preamble to the U.S. Constitution. Thanks to Schoolhouse Rock, the only hard part of the assignment was saying it without singing.
“The Math of Love Triangles” manages to get in some actual math instruction against all odds:
Murder By Numbers by The Police sounds like it would be good for reference, but the song only provides one way of bumping your victim off. (“If you can slip a tablet into someone’s coffee / Then it avoids an awful lot of mess”)
I was gonna say:
You don’t tug on Superman’s cape…
But then I remembered how the song ended. Slim didn’t learn that lesson.
“How to Save a Life” by the Fray is about how to fuck up an intervention, so you could probably reverse-engineer a good one from it. (Another heartbreaking song.)
Solidarity Forever, but you need all 6 verses.
In Sunday School we were taught a song to learn the books of the New Testament. It was a bastardization of Twinkle Twinkle Little Star. I would probably still be able to sing it upon request.
Jonathan Coulton: The Presidents
It only goes up to 2005, though.
Which produced several parodies by Axis of Awesome; “How To Bake a Scone”, “How To Catch a Duck”, and “How To Kill a Hooker”.
The song “Signs” teaches you how to get shot climbing over a fence onto what is clearly marked private property.
The Epic Rap Battles of History are surprisingly informative most of the time. I didn’t really know who Mansa Musa was until he was featured in an ERB.
Other educational songs by Jonathan Coulton include “Mandelbrot Set” (Just take a point called z in a complex plane, let z1=z^2+2, and z2=z1^2+z, and z3=z2^2+z, and so on, if the series of z’s will always stay, close to z and never trend away, that point is in the Mandelbrot Set!) and “Kennesaw Mountain Landis” (A fanciful retelling of the Black Sox scandal in which the eponymous baseball commissioner is “seventeen feet tall and had a hundred and fifty wives” and defeats Joe Jackson by shooting his middle finger off with a sniper rifle while riding in a blimp).
Harry Chapin is a good example. He was a storyteller, who told stories through music. Sadly, most of his stories were too long to be played on commercial radio (“If you’re gonna have a hit, you gotta make it quick, so they cut it down to 3:05” as Billy Joel sang in “The Entertainer”). “Cats in the Cradle” was one that fit the time constraints of commercial radio, so it got airplay.
But Chapin had a vast body of work, a lot of which actually had a lesson. “Cats in the Cradle” is one such, but let’s take a look at “Flowers are Red,” which tells us how to destroy a child’s creativity. And by implication, how not to do so. “Taxi” and “Sequel” teach us that changes occur as we go through life, and what we knew when we were in our twenties, is not what we will experience in our forties. Chapin has many examples:
“Mr. Tanner”: Don’t get too full of yourself. or let others get you that way.
“A Better Place to Be”: Maybe what you’re looking for has been here all the time.
“Bummer”: Don’t judge a book by its cover, and give veterans a break.
None of these songs would fit commercial radio, yet they each have a lesson. To be honest, not every Chapin song did (is “Thirty Thousand Pounds of Bananas” anything but a foolery song?), but the point remains: many of Chapin’s story songs, long as they are, contain lessons.
The Animaniacs show had several. One was all the nations of the world, another was the 50 states and their capitals. They had one on the planets, one on the presidents, once on the ingredients in ice cream, and one on Magellan and his voyage.
Right off the top of my head. I’m sure there are others.
And in a cruel twist of fate, half the world thinks it’s by Cat Stevens, both out of confusion with Stevens’ song “Father and Son” and because it has the word “cat” in it.
You never count your money while you’re sitting at the table.
But perhaps more to the point saying with one needs to know is a lesson in itself, what skills they need to acquire.
Sure, this song has a lesson, too. Truckers should be sure to check their brakes before starting down long steep grades and should also shift to low gear, especially when there is a sign telling them to do so. 
And personally, I don’t drive a truck, but the lyrics flash through my head every time I find myself going 90 mph (typically briefly when trying to pass someone on the highway).
In the live version, there’s also a lesson in how to properly end a song. 
Weird Al’s “Pancreas” teaches both anatomy and physics:
Apparently spitting into the wind is also ill-advised - and you really don’t need to see what the Lone Ranger’s real face looks like.