Superman's radio serial started in February 1940, and I have listened to the first eight episodes. The types of stories told hew quite closely to the comics, but there are some significant deviations that I'll mention later.
The first episode is set on Krypton, and tells the familiar story of Jor-L and Lara trying to save their people from the imminent destruction of their planet. It's very similar to the version of the story that appeared in the newspaper strip. It probably has the worst voice acting of the episodes I have listened to, but that's understandable, because these characters aren't going to appear again. There's a lot of bellowing, and declaiming, and it's all rather stilted and lacking in emotion. But for all that, it's quite striking to hear it dramatised. I don't think I ever realised exactly how grim Superman's origin story is until I listened to this.
Episode two is where the story deviates significantly from the comics. Little Kal-L makes his rocket trip to Earth, but instead of landing as a baby and being found by a passing motorist, he is fully grown by the time he reaches our planet. Ma and Pa Kent are nowhere to be seen. Superman saves a man and his son from a car crash, and they help him decide how to acclimate to Earth. They give him the name Clark Kent and suggest that he get a job as a newspaper reporter. I'm guessing here that the creators of the show wanted to get into the action quicker, and so dispensed with Superman's childhood, but I feel like a vital part of the story is missing here. Superman's desire to do good is just there, rather than being the product of a good upbringing.
When the newly named Kent goes to get a job as a reporter, he goes to the Daily Planet, and the editor there is not George Taylor from the comics but the much more recognisable Perry White (at least to modern audiences). Both of them are cut from the same "gruff editor" template, so I'm not sure why there was a change. I guess someone just disliked the original name.
The rest of the episodes deal with a villain known as the Wolf, who is making trains disappear under the orders of a mastermind known as the Yellow Mask. I didn't find it particularly enthralling. The most enjoyment I got was from the tidbits of Superman lore that kept cropping up. The opening, with the "It's a bird! It's a plane!" lines was an obvious one that sent a chill up my spine to hear it in its original appearance. Another unexpected one came in episode 6 or so, when Superman shouted "Up! Up! And away!" just before taking off. And that reminds me of something else: Superman seems to be flying here. In the comics at this time he still jumps everywhere, but although the radio show hasn't explicitly stated it yet, it has Superman doing things in the sky that sound a lot more like flight.
If you want to listen to these they can be found here: http://www.archive.org/details/superman_otr. They're an enjoyable bit of historical curiosity.