IMHO, the Showa Godzilla Series is chock full of memorable moments (so much that we even get countless model kit dioramas of them in Japan)! It's hard to pick just one.
I do agree that Mothra Vs. Godzilla, often hailed as best film by the majority of fans next to the original, has some of the greatest moments of any film. Godzilla's first appearance (called the best entrance in all the films), his smashing the castle in Nagoya, the US-only "Frontier Missile" sequence not seen in Japan (aside from a brief shot of it in the Japanese trailer), his encounter with the adult Mothra, his stumbling upon the electrocution towers, one of the Mothra larvae biting Godzilla on the tail, etc., etc.
Even Ghidorah the Three-Headed Monster has great moments as well, such as the title monster's first appearance, Godzilla and Rodan's humorous but mind-blowing battle, and the all-out climactic battle with all four monsters!
Among the Showa films, the 70s films have my personal favorite moments, notably including Godzilla Vs. Megalon (including the two handshakes between the heroes Godzilla and Jet Jaguar, which is personal to me, as it was like seeing two of my favorite character types, a dinosaur-like monster and a giant robot superhero, working together as friends and heroes). I'd say that may be my favorite moment in a Godzilla movie.
With the VS Series, I'd say Godzilla 84 and Godzilla Vs. Mechagodzilla '93 have the best moments in those films (the former for the FX scenes, and the latter for Godzilla finally winning).
With the Millennium Series, it was GMK and Godzilla Final Wars (the former for the fight between Godzilla and Baragon, among other scenes, and the latter for the best moment in the film, Godzilla Vs. GINO)!