Superman 113 (May 1957) is the first issue of the book to have a full length story, by Hamilton, Boring and Kaye.
A kryptonite meteor lands on Earth, and it contains a helmet, which is a recording device used by Jor-El. Wearing it, Superman learns the story of when his father acted as a hero to save Krypton from an invasion by an alien race, the Vergoans. Jor-El heads out to the asteroid the aliens are using a base, and discovers that, off his planet, he has super-powers.
This story is also the first time that Superman’s Kryptonian name is revealed, Kal-El. Although his super-memory allowed him to remember the flight to Earth, he did not recall what his parents called him, apparently.
The story gets interrupted as giant plants begin destroying Metropolis. A mad scientist is behind this, but the point of the interruption is to allow Superman some action – and he uses the same technique he had just learned that his father had used to protect Krypton.
Jor-El learns that the Vergoans were planning to destroy Krypton, using the released energy to power their failing sun. The queen of the Vergians, Latora, explains that they chose Krypton because it was doomed to destruction anyway. According to this story, that is how Jor-El discovered that Krypton was going to be destroyed.
Back in the present, Superman defeats the mad scientist on the loose, and then seeks out the Vergoans. They are still hunting for a way to power their sun, and Superman takes care of that for them. Latora, who looks much the same as she did in the flashbacks, despite the twenty some year difference, really ought to be the villain of the story, but Superman treats her as a damsel in distress anyway.
Probably because of the detail of Jor-El learning about Krypton’s doom from Latora, this story was ignored by later continuity. The other problem was the notion of Kryptonians gaining powers simply by leaving their planet. Once the idea of their powers coming from the yellow sun were introduced, this story became “wrong.”