I'll throw my hat in this ring, since Pokemon is most of my childhood and still a large part of my newfound adulthood...
Technically first encountered the shiny red Gyarados in Silver and Crystal. That one is scripted. It doesn't count.
Actually first caught a shiny geodude in LeafGreen's victory road. It's been lost to the aether though, must've deleted my save file with it when I was a kid, I didn't have a way to trade it between gen 3 games.
I don't think I encountered any shinies in gen 4.
The next I remember for sure was shiny Basculin in White 2. I was EV training at the time. And now I absolutely hate Basculin because (A) it's a terrible Pokemon, (B) super generic design, even with the two forms, and (C) its shiny is virtually no different than normal. To encounter something so rare but so underwhelming still upsets me.
I think there was one in generation 6 but I can't recall specifically.
In moon I found a shiny Hakomo-o. I was SOS chaining to find one holding a razor claw (because that was the only way to get one, and get Weavile by association) when it spawned, I remember saying "forget the item, I'm catching the shiny!" Turns out it was also holding a razor claw. So, hands-down that was the best shiny encounter.
In Ultra Sun I found a shiny Staryu and shiny Pineco. No neat story with those though, although I think I was SOS chaining for competitively viable pokemon with both cases.
And a friend has given me a few shinies. However this friend also hacks / generates Pokemon for competitive play. Personally I'm fine with hacking like that, but I can't say in good faith those shinies are legit.