I'm a pretty big fan of Godzilla, I've seen every film and own most of them on DVD/VHS. Here are my top 5 movies:
#5 Godzilla vs. Megalon
Was supposed to be a solo movie for Jet Jaguar, but Godzilla and Gigan were added since Toho didn't believe JJ could sell a movie himself. It is a pretty shitty movie overall, overuse of stock footage, terrible editing, but I still really enjoy it because Jet Jaguar is such a likable kaiju.
#4 Godzilla vs. Mechagodzilla
This film is stellar, the best of the Shōwa Era by far, the fight scenes are incredible and the mystery behind the fake Godzilla in the beginning keeps the viewer intrigued. King Caesar makes his first appearance and is another fantastic kaiju, one of my favorites. Caesar's fight with Mechagodzilla is hilarious and like Jet Jaguar, he is an incredibly likable monster.
#3 Godzilla Against Mechagodzilla
Going to see a theme here with my favorites, but Mechagodzilla/Kiryu is my favorite kaiju. This probably has one of the best plots in a Godzilla film (which isn't really saying much) and the fights are some of the best. Incredible entry in the series and one of the more well rounded films that even a non-kaiju fan could enjoy.
#2 Godzilla: Tokyo S.O.S.
Sequel to #3, S.O.S. begins where GAM left off. The addition of Mothra makes for a compelling story and I think this film has THE best action in the Godzilla series. It has a touching end, and if this was the final entry in the Godzilla franchise it would've been absolutely perfect, but alas when you have a cash cow you keep making films.
#1 King Kong vs. Godzilla
I'm pretty sure this is the first Godzilla film I saw and owned. The two biggest names in monster movie history duking it out made for a pretty compelling story. The English version is a bit different and despite being shorter, is just as good. The ambiguous ending was pretty fitting for such a clash, shame these two monsters will probably never meet again. Also happens to be one of my favorite movies of all time.
Honorable mentions:
Godzilla vs. Gigan
Godzilla vs. Destroyah
The Return of Godzilla
Godzilla vs. SpaceGodzilla