My very first shiny that I encountered was a Shiny Tentacool when I was surfing between Cianwood and Olivine City in Pokemon Gold. At the time I really didn't pay attention to it because to me, it was just another Tentacool so I ran from it, trying to swim as fast as possible because I hated surfing at the time.
Didn't find out until a few years later, that it was special.
My first actual Shiny that I caught was in Pokemon Heart Gold. I was training on Mt. Silver to face off against red and I ended up encountering a Shiny Ponyta. It was my very first shiny and made me love Ponyta as one of my favorites.
Not long after I ran into a Shiny Hoppip in the Safari Zone in HG, but ended up trading it for a Legendary in the GTS because I really didn't care for it. Hoppip may have actually been first, but I always forget about it, so I consider Ponyta my first, as it may still have been my first since I didn't really play around with the Safari Zone in HG until much later.
But that Ponyta really started getting me into Shiny Hunting.