Showa Era
The Showa series of Godzilla films are the orignal classics. Starting with the 1954 original, this series spanned until the 15th Godzilla film in 1975. This series is best known for Godzilla the bad-guy-turned-good resulting in some very classic movie monster cheese.
Whether the later films are an insult to what the original film was trying to say is up to the viewer to decide. Obviously it's not a popular opinion, otherwise we wouldn't have seen as many sequels as we have.
Godzilla, King of the Monsters (Gojira, 1954)
Godzilla Raids Again (Gojira no gyakushu, 1955)
King Kong vs. Godzilla (Kingu Kongu tai Gojira, 1962)
Godzilla vs. Mothra (Mosura tai Gojira, 1964)
Ghidrah, the Three Headed Monster (San daikaiju: Chikyu saidai no kessen, 1964)
Godzilla vs. Monster Zero (Kaiju daisenso, 1965)
Godzilla vs. the Sea Monster (Gojira, Ebira, Mosura: Nankai no diaketto, 1966)
Son of Godzilla (Kaijuto no kessen: Gojira no musuko, 1967)
Destroy All Monsters (Kaiju soshingeki, 1968)
Godzilla's Revenge (Gojira-Minira-Gabara: Oru kaiju diashingeki, 1969)
Godzilla vs. Hedorah (Gojira tai Hedora, 1971)
Godzilla vs. Gigan (Chikyuko kogeki meirei: Gojira tai Gaigan, 1972)
Godzilla vs. Megalon (Gojira tai Megaro, 1973)
Godzilla vs. Mechagodzilla (Gojira tai Mekagojira, 1974)
Terror of Mechagodzilla (Mekagojira no gyakushu, 1975)