In "The Super-Tests of the Super-Pets" we don't exactly find out how a dog, a cat, a monkey and a horse who used to be a centaur learned to work together, because this story from Adventure Comics #322 is basically about much they DON'T want to let anything else join the Legion of Super-Pets. Xenophobia is so rampant here, it's hard to believe it comes from an organization that features four (maybe five) different species.
It all begins when the Legion prepares to go fight the Time Trapper and have to leave someone in charge of defending the Earth. That's to be the LSP, and Proty, Chameleon Boy's blobby "pet", after being denied the possibility of going on the trip, tries to join the interim team. His application isn't well received.
-Comet: Proty must impersonate Superboy (he doesn't gain his powers) and accept his next mission whatever it is (Proty has to go to another planet and recover some important crystals).
-Streaky: Proty must fool Saturn Girl, a telepath, while impersonating another Legionnaire.
-Super-Monkey: Proty must prevent the Legionnaires from entering the club-house for a whole hour (even though this is a state of emergency with everyone preparing for temporal war).
Proty succeeds every step of the way, and nobody hits any of the Super-Pets with a rolled-up newspaper for screwing with him. One test left. Krypto's. And here's a clue as to how that'll go.
Around this same time... in Adventure Comics #320, Superboy remembers the fun he had with his pup and a robot dog when he was a child.