Reddit mentions of Godzilla (Blu-ray)

If you want the best picture quality, absolutely go for Blu-ray. If DVD is good enough for you, then by all means go for it -- it's generally cheaper anyway. Some films are available in multi-format combo packs; if you're fine with DVD for now but might switch to Blu later, then it might make sense to grab those.

The rights are definitely all over the place. Here's a handy breakdown of who owns what. Those Blu-ray double features (plus Godzilla 2000) cover the Toho-produced films from the '90s and '00s.

Criterion has released an excellent DVD and Blu-ray of the original Godzilla from 1954, which is an absolute must. Other than that, as far as we know, they currently own and are apparently working on discs of the rest of the Showa Godzilla series except for King Kong vs. Godzilla, Godzilla vs. the Sea Monster, Godzilla vs. Hedorah, and Godzilla vs. Gigan. (They also own Rodan and War of the Gargantuas; no word on whether those specifically will be released on disc.) We don't yet have release dates or official announcements for any of these films, and I personally wouldn't expect to hear anything before February, which is when their May 2019 releases will be announced.

If you just want to see as many of the films as possible and worry about the best available release of a given film later, here's a shopping list:

• Classic Media's Godzilla Collection box set. A good chunk of the Showa series in Japanese with subs and dubbed. OOP, but can probably be found cheaper than all the individual releases.
  
• Both volumes of Sony's Toho Godzilla Collection. Covers the '90s-'00s Japanese films, plus Rebirth of Mothra I and II. Note that these are reissues of older discs, so some are dub-only and/or pan and scan.
  
• Godzilla Stomp Box, containing Godzilla vs. Megalon and Destroy All Monsters.
  
• King Kong vs. Godzilla, Godzilla vs. the Sea Monster, Godzilla vs. Hedorah, Godzilla vs. Gigan, The Return of Godzilla, Godzilla vs. Biollante (OOP), the 1998 and 2014 American films, and Shin Godzilla are all available in standalone releases.
  
• The anime films are on Netflix. No word on whether these will get physical releases.
  
  Of course, aside from physical media, there are always streaming options of varying legality as well as bootleg discs, but I doubt the mods would take too kindly to my posting specifics about those here.
  
  Hope this helps!

Amazon has a pretty good deal on the twin pack