You can check out the previous installments of this rather lengthy series here.
Action Comics #591
If the cover isn’t clear enough for you, this issue has Superman fighting Superboy while the Legion of Superheroes watch from a safe distance and bet on which one’s going to come out on top. The issue also has Krypto the super dog, and some semblance of comic book continuity. In the interest of saving you a lot of time, don’t bother wondering who won. I read the issue five times, used an excel spreadsheet to score each combatant, and I’m still not entirely sure who won.
As for why they are fighting, Superman #8 had Superman learning of a parallel reality where Clark Kent already had fully developed powers as a kid, became Superboy, and traveled to the future and joined the Legion of Super-Heroes. It turns out this was all part of an extremely roundabout ruse by a villain named The Time Trapper. He’s a guy who lives at the end of time and spends all day watching costumed teenagers through his time portals, apparently. TTT is so obsessed with those crazy kids at the Legion of Super-Heroes that he created an entire universe where Superboy exists and then had the Boy of Steel infiltrate the team. There was probably a less complicated way to get a spy in there, but you have to admire the guy’s insanity/dedication.
Anyway, Superman ends up in Superboy’s universe and meets alternate versions of his friends and family in Smallville, then gets into a fight with Superboy (who is under the control of The Time Trapper) and his intelligent flying dog Krypto. However, Superman then realizes Superboy is intentionally throwing the fight because he doesn’t want to obey the Traper’s orders anymore. So, Superboy turns good again and he and the Legion of Super-Heroes go off to fight The Time Trapper at the end of time, but they don’t let Superman come with them because he’s old and lame. (And to protect the sanctity of the timestream I guess.)