I've done a lot of research and web surfing, and found what seems to be a good breakdown of the continuity of the Godzilla movies. Now, let's bear in mind: Not all of the movies are canon to one another. A lot of the movies break away and do their own thing. The intel found in the article below pretty much matches up with everything else I have found online in regards to the subject, though I will add that the American Godzilla is referenced in GMK, and as such can be considered canon canon with that film.
The original article was found here: http://www.godzillatemple.com/movie_notes.htm
Original Series (1954-1974)
There has been a lot of discussion lately regarding how the different Godzilla movies relate to each other, and events in certain movies have been used to help "prove" various theories about Godzilla. For example, the treatment of the title monster in "Gigantis, The Fire Monster" as a separate monster than the one in "Godzilla, King of the Monsters" has led some people to argue that the Godzilla portrayed in all the movies after the first one was, in fact Godzilla's mate, and is therefore female.
So, what's the truth? Beats me! What's my opinion on the subject? Read on ...
Essentially, the early Godzilla movies weren't attempting to achieve any sort of coherent continuity among the various films. The first movie was meant as a one-shot "movie with a moral", and Toho was as surprised as anyone when the movie did well enough to warrant a sequel. Although the second movie, "Gigantis, The Fire Monster", did make reference to the first movie, it claimed that both monsters were members of a species of "fire monsters" which had been around since prehistoric times, whereas the first movie indicated that Godzilla was created through man's testing of atomic weapons.
In the next movie, "King Kong vs. Godzilla", Godzilla was treated as if no one had ever seen him before. Instead, he was believed to be a "prehistoric species of dinosaur" who had been lying dormant, frozen in an iceberg, for almost 100 million years.
I've always liked to think of there being only one Godzilla for all the original movies, with the thought that somehow he regenerated from his "death" in the first movie. Basically, if there were more than one of him around, why would only one show up at a time? Why would they take turns attacking Tokyo? Having said that, though, he did disintegrate pretty thoroughly at the end of the first movie, so maybe this is just wishful thinking on my part.
"Hesiei" Series (1984-1995)
With the release of "Godzilla 1984" (a.k.a. "Godzilla 1985"), Toho made a conscious decision to rewrite Godzilla history. All the movies after the original were deemed to have never happened at all. Thus, any discussion of the later Godzilla being the original Godzilla's mate appear to be mooted.
For a while, the new series attempted to follow a strict continuity, with each movie picking up where the previous one left off, and with frequent references being made to previous films. Thus, at the end of "Godzilla 1984", Godzilla falls into a volcano, and in "Godzilla vs. Biollante", Godzilla arises from the same volcano. Similarly, in "Godzilla vs. Biollante" Biollante heads off into space, and in "Godzilla vs. Space Godzilla" there is speculation that Space Godzilla could have been created from some of Biollante's cells carried into space. In "Godzilla vs. Mechagodzilla", it is revealed that Mechagodzilla was built using technology gleaned from the cybernetic version of King Ghidorah from "Godzilla vs. King Ghidorah". Finally, "Godzilla vs, Destroyer" comes full circle and has a monster created from the device which originally defeated Godzilla in the first Godzilla movie.
All of this carefully maintained continuity, however, falls apart with the events depicted in "Godzilla vs. King Ghidorah". In that film, time-travelers remove Godzilla from the island where he was exposed to the nuclear bomb test which turned him into everyone's favorite radioactive reptile in the first place. Unfortunately for the time-travelers, Godzilla lies dormant in the ocean near Tokyo until he is exposed, many years later, to the radiation given off by a sunken nuclear submarine. This means that, not only did the original "Godzilla, King of the Monsters" never take place, but neither did the events portrayed in "Godzilla 1985" or "Godzilla vs. Biollante"! And if these movies never occurred, then where did Space Godzilla and Destroyer come from?
Anyway, the point is that the movies that make up the so-called "Heisei" series [from "Godzilla 1984" through "Godzilla vs. Destroyer"] did follow a loose continuity that was unfortunately mucked up royally by the time-travel events in "Godzilla vs. King Ghiodrah."
"Alternate Reality" Series (1999 -2001)
When Toho restarted the series for a third time with Godzilla 2000, they threw the Heisei continuity out the window and started again from scratch the same way they did with "Godzilla 1984." The main difference, however, is that they decided to not have any continuity whatsoever among "Godzilla 2000" and two movies that followed it. Which is to say that Godzilla 2000, Godzilla vs. Megaguiras, and GMK: All Monsters Attack are each meant be stand-alone films, each retelling the legend of Godzilla from a new perspective. For this reason, these three films are commonly referred to as the "Alternate Universe" series, since each film created its own unique Godzilla universe.
New Generation Series (2002 - 2003)
And then, in 2002, Toho came out with Godzilla x Mechagodzilla (III), which once again started the whole thing over again from scratch, recognizing only the original movie as part of its continuity. The following movie, Godzilla vs. Mothra vs. Mechagodzilla: Tokyo S.O.S. is a direct sequel to Godzilla x Mechagodzilla (III). Since these movies have shared continuity, they cannot properly be called part of the "Alternate Universe" series. Therefore, for lack of a better term, I am currently referring to them as the "New Generation" movies.
Godzilla: Final Wars (2004)
Finally (for now, at least), we have Godzilla: Final Wars. Once again, continuity is thrown to the wind and the series is started over again for the purpose of finishing it. Well, sort of. The movie is certainly not a direct sequel to the previous two films, and there's no mention of the events that took place in those films. However, the movie is designed to be the culmination of 50 years worth of Godzilla films, and there are various references to many of the movies, both old and new, throughout. Basically, it's as if Toho took every Godzilla movie made over the last 50 years, stuck them together in a blender, hit "puree" and then threw the entire mess on the screen. Not that this is a bad thing, mind you....