Gen. II - Golbat
I found my very first non-default Shiny in Silver Version's Ice Path. I didn't know about Shiny Pokémon at the time and couldn't tell a color difference due to the GBC's dull display, so I thought my eyes were just playing tricks on me when I saw the sparkles. I tried to run from it, but I saw a "You don't want to do that!" message (I don't think that message is actually programmed into the games, so maybe my eyes actually were playing tricks on me that time). I caught it and later checked its stats in the PC and saw the three stars in the corner of the screen. I later evolved it into Crobat and it really weirded me out that it turned pink (I'm actually fine with pink Crobat now) and it looked kind of creepy in Pokémon Stadium 2, but at least I was able to rub it in my older cousin's face because he had never found a Shiny. :P
Gen. III - Nidoran M
I found a couple more Shinies in Gen. II before this, but my first Gen. III Shiny was a Nidoran (Male) that I found in FireRed's Safari Zone. I was so scared that it would run away! I didn't even bother throwing rocks or bait, I just chucked Safari Balls at it. The first one didn't work, which gave me quite a scare, but it didn't run away and I was able to catch it after one or two more attempts. I evolved it into a really cool blue Nidoking and made it part of my main team. Unfortunately, I let a friend borrow the game and they lost it. :(
Gen. IV - Gastly
My first Gen. IV Shiny was a blue Gastly in Pearl Version's Old Chateau. I had just learned about EV-training and was attempting to raise something in Special Attack when it appeared. I caught it and evolved it into Haunter and, with a friend's help, into Gengar. Had I known what Gengar's Shiny looked like, I wouldn't have evolved it. Because I didn't like its colors, I later traded it to someone on here for a most-likely RNG'd Shiny. I kind of regret trading it now, since Shiny Mega Gengar looks awesome, but you live and you learn I guess. =w= (Fun fact: Every Shiny I found on Pearl Version evolves into its final form via trade -- Gastly, Graveler, and Machoke.)
Gen. V - Basculin R
The first Gen. that my first Shiny wasn't a Poison-type. I found my first Gen. V Shiny while fishing for Basculin, EV-training in Speed. I battled a lot of Basculin on Black and Black 2! I kind of hate Basculin, but I had a thing for using any Shiny I found, so I caught and trained her. I named her Marge (probably the first Shiny I ever nicknamed). I later transferred her to X and found 3 more Shiny Basculin Red-Stripe, but she's still more special since Shinies were a lot harder to find before X/Y.
Gen. VI - Spheal
My first Gen. VI Shiny was a Spheal from the Friend Safari. I was actually trying to Masuda a Shiny Bergmite (which I eventually succeeded at), but I was getting a bit burned out on hatching eggs and decided to run around there for a while. I can't even remember if I knew about the increased Shiny rate at that point... Since then, I've probably found 30+ Shinies in the Friend Safari alone.