Reddit mentions of Godzilla Vs. Destroyah / Godzilla Vs. Spacegodzilla / Godzilla Vs. King Ghidorah / Godzilla Vs. Mothra (1992) / Rebirth of Mothra / Rebirth of Mothra II - Set

A few films I would check out:

• Terror of MechaGodzilla is one of the best Showa films. It has an amazing soundtrack, and has an interesting story and a cool 1v2 battle with Godzilla, MechaGodzilla, and the newcomer, Titanosaurus, who has an amazing roar.
  
  
• Godzilla vs Biollante is one of my favorite as well. Biollante may look like an abomination, but they make the creature seem "beautiful" in a sense, maybe precious is the better word. The story is very good, and the Biollante "suit" is really convincing even to this day. The human characters are interesting as well.
  
  
• Godzilla vs Destoroyah is...really, really good. I wouldn't watch this one first, leave this one as one of the last ones you want to see. Very emotional story, very dark, and not really uplifting in the slightest.
  
  
• GMK All Monsters All-Out Attack: Fantastic film. Godzilla is the villian this time around, while King Ghidorah, typically a villian, is considered a good kaiju. Good effects, a scary Godzilla, and a well paced story.
  
  
• Godzilla 2014: A good film to start off with. It has its strengths and weaknesses, but it's a good start regardless.
  
  
• Godzilla 1954: Obviously
  
  Films I would avoid:
  
  
• Godzilla vs SpaceGodzilla: This film has horrible pacing issues and a story that isn't interesting. While I found SpaceGodzilla interesting in the games, he isn't interesting at all in the film.
  
  
• Godzilla vs the Sea Monster: This movie is just...meh. I sometimes forget it exists.
  
  
• Godzilla vs Megaguirus: Another meh film. Megaguirus is a cool kaiju, but it just doesn't do anything for me.
  
  Films that you should watch with friends for some laughs:
  
  
• Godzilla All Monsters Attack: This film is all over the place. It really feels like a film that had a production team with a motto of "Hey, why not? It's not like we're trying"! It's hilarious on how weird it is.
  
  
• Godzilla 1998: Yeah, this film is basically nothing but the usual 90s cliches we've seen before, now with more Godzilla, sorry, Zilla. Most of the Simpsons cast is in it though, so that's a plus...right?
  
  
• Godzilla Final Wars: I don't think I should say much for this one. The craziest Godzilla film you will ever watch.
  
  
• King Kong vs Godzilla is great for the sole fact it's King Kong vs Godzilla, East vs West. It's not the best in terms of story, but it's cool to see both of them on screen together for the first...and not last...time.
  
  If you need help finding these films, here are some links for some packs on amazon that can fill up your collection pretty quickly.
  
  
• Godzilla Collection (https://www.amazon.com/Godzilla-Collection-Vol-Yuriko-Hoshi/dp/B000VXWK86/ref=sr_1_6?s=movies-tv&ie=UTF8&qid=1468168538&sr=1-6&keywords=godzilla)
  
  
• The entire Millennium series, with Godzilla vs MechaGodzilla 2 (https://www.amazon.com/Godzilla-Collection-Vol-Yuriko-Hoshi/dp/B000VXWK86/ref=sr_1_6?s=movies-tv&ie=UTF8&qid=1468168538&sr=1-6&keywords=godzilla)
  
  
• Some Hensei films, and Mothra films (https://www.amazon.com/Godzilla-Destroyah-Spacegodzilla-Ghidorah-Rebirth/dp/B013P6PEUG/ref=sr_1_12?s=movies-tv&ie=UTF8&qid=1468168538&sr=1-12&keywords=godzilla)
  
  I hope this helped in some way. Have fun watching them. If you have anymore questions, don't hesitate to ask.
  
  EDIT: I should add that Godzilla vs Gigan, and Godzilla vs Megalon are two films that are great to watch with friends as well.
  
  EDIT 2: Heck, watch all the Godzilla films with your friends, each one has something special to enjoy.

I think this and this are a great way to get your collection started. If you are looking to do Blu-Ray these double feature bundles are good too. Then just hunt down the rest. This one has a lot of the earlier ones. I've purchased individually because the sets weren't always around, I'm thinking of upgrading to blu-ray but i still love running across the ones I'm missing in places like thrift shops. Anyways. Good luck.

Edits. Typos