Mine was Bellsprout in 4th gen, I remember it to this day. My reaction was basically 'omg shiny!!! but why does it have to be bellsprout of all pokemon', in a nutshell.
A Rattata in Crystal... I was so extremely excited when I found him xD I spammed pokeballs without attacking because I didn't want to accidentally killed him. Lucky I hoarded pokeballs like no tomorrow!
My first shiny was Wingull in Pokemon Ruby but I've never heard of shinies at the time (since Ruby was my very first Pokemon game I played) so I defeated it as usual. I didn't take notice of the green stripes on it's wings at the time x'D
My first Shiny Pokemon was a Rattata in Heart Gold, right outside the Daycare. I caught it, and gave it the stupidest name in the world (I named it "FirstShiny" [no space in between the two words, either]). I still have it, too, since I transferred it to Alpha Sapphire.
My first ever shiny pokemon was a shiny Wobbufet in Pokemon Y. I was looking through the friend safari, and I caught it. I almost killed it, but luckily I caught it on the 5/5 of my PokeBalls lol. I squealed so loud I woke up my dad.
My first legit shiny was a Purugly in Pearl. I was grinding on rt 221 & almost missed the sparkle it gave off. It's still sitting in Pearl, I've never used it for anything.