>>212929185 (OP)1954 Gojira and then Minus One to start. It bookends the franchise, gives you the original intended tone for the character that's kind of been lost through decades of flanderization, and pop culture reference and parody, and shows how far it's come in 70 years, plus Minus One is a soft remake anyway.
From there, I'd do Shin (which you'll also appreciate even more if you're familiar with Evangelion).
From that point, I'd go back to '84, and then run through the Heisei (90s) series, since those are narratively connected, and maintain a fairly consistent tone.
After that, do the Millennium series ('99-'04). They're a bit more confusing than Heisei, because they're all narratively disconnected (aside from x Mechagodzilla and Tokyo SOS). They also dip a bit more into the campiness than everything else on this list. I'm trying to ease you into the campier stuff, if you can't tell.
Then, If you're still into it, I'd wrap up with the Showa series. It's fun to see what inspired the stuff that you've already seen by now, and the more crude suits and effects won't be such a turn-off. I feel like a lot of people saw Minus One, decided to start watching the movies from the very beginning, and then bailed.
'98 and Monsterverse are completely up to you.