Not that I need another franchise to get into, but which one(s) should I watch next?
I agree with the Original/1954 Gojira. That makes sense to go next.
Shin Godzilla would be my next recommendation. It's very different than the other two. But, I think I can say that this is the last of the "deep" Godzilla movies. The ones that actually are going for a big metaphor during their run time. The Original, Minus One, and Shin are probably the 3 "deepest" Godzilla movies you can find.
After that, there's the fun ones. These are basically just going to be excuses to see cool monsters fight each other. Which has it's own charms.
Godzilla Final Wars is just pure spectacle. Godzilla fighting almost every single kaiju in his entire history. It might mean more once you get a few more movies under your belt. But, I do think this is probably the pinnacle of Godzilla as a popcorn flick.
Godzilla vs Destoroyah has a memorable enough enemy kaiju and ending that I think it's worth checking out. Even if you'd be skipping to the end of the Heisei Era's overarching story. (Which...isn't that much of a "story", really, don't worry about spoilers.)
Godzilla Against MechaGodzilla (2002) and Godzilla Tokyo SOS (2003)-This is the "Kiryu" MechaGodzilla duology.
Godzilla vs King Ghidorah (1991) is just a lot of fun.
Godzilla King of the Monsters (2019) is another big fight fest. You could start with the first Legendary Godzilla movie from 2014 if you want. But, I think KOTM is more fun. This would allow you to dip your toes into the "American" branch of recent movies. If you end up liking it, follow it up with the two Godzilla x Kong movies.
And I'm debating on which of the other Showa era movies would be the best to watch. I think I'm going to recommend Godzilla vs Hedorah. The enemy kaiju is memorable, and it's a little different than some of the other movies in that era.
That should give you a little sample of all the major eras of Godzilla. I think those are most of the best ones. Not all of them, but it might give you a idea of which era you want to see more from.