I became a Godzilla fan in my early years. My parents used to tape them on Sci-Fi and pick up copies at the flea market. Bottom line, I watched way too much Godzilla as a child and developed a taste for it.
As someone already pointed out, Son of Godzilla is one of the better movies. Nice pace, some funny moments, and one of the best Godzilla entrances in all of the movies. Godzilla Final Wars is great too, although a bit rough on the second play. You have Frye (in a weird role to begin with) and Godzilla...all the bits in between the two are annoying.
Finally, if you are wanting to see some really crazy shit check out Godzilla vs Hedora (might be called the Smog Monster in the U.S. releases) and Godzilla vs Megalon. The former tried to go back to Godzilla being a commentary on the environment and is just plain weird for the most part with crazy little cartoons mixed in with the movie. The latter is widely regarded as the worst movie in the franchise. I have a soft spot for it though as I have a penchant for screaming Easter islanders and Ultra-Man ripoffs (Jet Jaguar is a much better character than Ultra-Man anyways).
PS Godzilla 2000 isn't that bad. I remember watching the trailer for it in the way back and it had music by Powerman 5000 (I think). I really liked the Godzilla design in that movie.