The most prominent appearance of the Moon in the show is in "Planet Jackers". Along with Earth, it is stolen by the Planet Jackers in order to be thrown into their dying sun as fuel. When Zim and GIR cut Earth and the Moon out of the containers the Planet Jackers were carrying them in, the Moon briefly bumps into Earth's surface before drifting away, presumably back into its original orbit. Both are left drifting a different star than normal, but due to the series' negative continuity, they are restored to their normal place in the universe in all future episodes.
In "Tak: The Hideous New Girl" Zim revealed that he promised the moon to GIR.
In Issue 21, after Zim and Gaz switch bodies in an accident with one of Zim's machines, Gaz offers Zim a bet that she can conquer Earth more efficiently in three days than he has in years, with the loser to be banished to the Moon. Zim arrogantly agrees, thinking Gaz won't succeed. However, she does, and the issue ends with Zim now banished to the Moon, along with Dib, who likewise swapped bodies with GIR.
In Issue 32, Professor Membrane flies to the Moon, citing an unidentified situation there that he's needed for. This seemingly contradicts Issue 16's statement that he's unaware of travel to the Moon being possible (but presumably is just another case of the series' lack of consistent continuity).