Fpr the Showa series (the first group of Godzilla movies), watch Gojira (not Godzilla: King of the Monsters) through Mothra vs. Godzilla for the original "serious" Godzilla movies, although obviously it starts to get a bit sillier after the monster fighting is introduced. Gojira is a classic and the others are pretty good, and King Kong vs. Godzilla is just fun. Godzilla Raids Again is oft forgotten but pretty good I think. MAYBE throw in Ghidorah the Three-Headed Monster (the fifth movie) afterwards just to see the direction the franchise started going in, and because it's Ghidorah. If you want to watch the goofy campy Godzilla movies, they start here.
Then skip ahead to Godzilla vs. MechaGodzilla just because it's practically required watching.
For the Heisei series (the second Godzilla series, ignores everything after the original Gojira and picks up from there), start with The Return of Godzilla and Godzilla vs. Biollante, maybe Godzilla vs. King Ghidora, but the others can be skipped. Godzilla vs. Destoroyah perhaps as well because it's an important point in the franchise.
Skip the American Godzilla. Not a bad monster movie but it's not Godzilla.
For the Millennium series, just watch Godzilla, Mothra, and King Ghidorah: Giant Monsters All-Out Attack, which is BY FAR the best Godzilla movie since the original. None of the other Millennium movies compare, so don't bother.