Most of the running gags with the big-reared Galactic Grunt. Lucario nonchalontly poking the guy with the spike on his paw has to be one of the funniest moments
Most of the back pages of the books have a random Pokémon pointing out you're reading the book the wrong way, and Hareta translating in the form of "[Pokemon] says you're reading in the wrong direction!". Volume 5's example? An insane Cyrus glaring and growling at you. "Cyrus says you're reading in the wrong direction!"
Volume 6 also has Hareta tell the reader that his dad is saying you're reading it wrong, and his dad's reaction is something along the lines of "Quiet son! You're going to ruin my surprise appearance!"
Mitsumi and Mars going gaga over Minun in Volume 7.
Hareta calling Fantina 'Grandma', to her annoyance.
Cynthia sitting on a Galactic Grunt while Fantina's laughing her ass off over it in Volume 8.
Mitsumi claiming to have gotten lost due to forgetting how the warp panels connect in the Galactic Hideout in one the later volumes. Pretty funny whether or not she's telling the truth, as no one seems to believe her.
Jun yelling at Palmer for trying to flirt with Mitsumi.