My eyes have been opened to Toho!

If you haven't watched the original 1954 Godzilla film, do so immediately. 

After that, you can either watch them in release order or just pick whatever titles strike your fancy. There are other Toho monsters for sure, but the only one with multiple franchises is Mothra. 

Movies that are a must watch include the following.

Gojira 1954

Shin Godzilla 2016

Invasion of Astro Monster 1965

Mothra vs Godzilla 1964

Ghidorah The Three Headed Monster 1964

Terror of MechaGodzilla 1975

Godzilla vs MechaGodzilla 1974

King Kong vs Godzilla 1962

Godzilla, Mothra King Ghidorah Giant Monsters All Out Attack 2001

Godzilla 1984

Godzilla vs Biollante

Godzilla vs The Sea Monster 1966

Gamera Guardian of the Universe 1995

Gamera 2: Advent of Legion 1996

Gamera 3: The Revenge of Iris 1999

Pacific Rim 2013.

These films are top notch filmmaking and are quite honestly, the best quality of Kaiju films. I'd definitely recommend these as these are all highly praised films by fans and critics alike.

Just watch all of them.

Gotta take the good WITH the bad, or else we'd never learn :P

The Toho movies were pretty solid. But honestly, I watched a Godzilla Halloween marathon a couple years ago and they started to blend together. It got to a point where I seemed to perk up when I saw the Toho logo at the start of the movie. 

Can't remember the movie, but there was one with subtitles. It was really  good but frustrating trying to read and watch at the same time. 

   Rodan 1956

Shin Gojira (2016) good for new Godzilla fans but also keeping that old style of Godzilla.

Watch Ghidorah the Three Headed Monster. It's one of the best, and the first movie featuring Godzilla, Rodan, and Mothra teaming up. Even if it's just a Mothra Larva, it's still a good movie.

See, this is exactly what I mean, when you make a film that appeals to the general audience, a film that everyone can watch, not only do you gain more revenue, but you gain new fans as well.
THIS is why I love the MonsterVerse so far, it does exactly the same thing as Marvel: it acknowledges the fans and shows respect for the source material while still making something fresh and new that general audience will like. I prefer they keep it like that and not make the same mistake DC had made by catering the fan bases only at the expense of the bigger audience.

if you want serious shin godzilla, and the 1954 godzilla film. if you watch fun films, godzilla vs mechagodzilla, godzilla final wars. others films that dont include in here that i would recomend would be the 90's gamera trilogy, pacific rim, theres two anime kaiju seires that i recommend, ssss gridman and the new ultraman series. Finally i would recommend any obscured b movie that not that many talk about as alot have their charms.  

That's why I love the new movies because it makes new fans to the fanbase and makes new characters like the MUTOS and other kaiju but my wish is toho does not sell it off again unless back to legendary then I'm fine with it . Just hopefully DISNEY stays off of toho they already took marvel leave big g alone 

I definitely recommend all the Mothra movies.   My favorite Godzilla movies are

Godzilla Tokyo SOS

Godzilla Against Mechagodzilla

Godzilla 1954

Godzilla vs Destroyah

But really,  I enjoyed all Godzilla movies.  These are just the ones I can rewatch all day.    I honestly have only seen one Gamera movie. But I like his origin and would like to see more. I do recommend Gamera. 

I would like to see Godzilla junior return including a Godzilla vs Gamera 

besides being a fan for years and still looking for some of the films on dvd