I can't remember exactly what my reaction was with my first ever shiny, a shiny Gulpin, so I'll just tell you a few- of the two shinies I failed, then the first one I caught and kept.
It actually happened while I was playing through Pokemon FireRed (and actually bothering to complete it for the first time). I was in the Safari Zone, looking for the area in which you could catch Scyther, when a shiny male Nidoran appeared. I was in shock- I just threw something... then it fled. Honestly, I was in shock so much that I don't even really think it hit me that I found a shiny. All I remember doing is just shrugging and saying shakily to myself 'Well, I have a perfectly good Nidoking anyway. I wouldn't have used it' and just getting on with my life. The second time, I wasn't so pragmatic about it. It was a shiny Fearow. Excited, I went to weaken it with my weakest move- only for it to crit and kill it in one hit. Needless to say- I was ticked beyond belief. It actually was a catalyst for my irrational hatred for the line. I managed to reclaim it recently, though, while playing through my SoulSilver Nuzlocke. Osiris, the one I caught was on my final team. (And, before you ask, yes, Osiris was awesome. I was just as stunned to find him, and as stunned to realise exactly how fun Fearow was to use).
I didn't find a new shiny until Y, though. While I was playing through it, I was trying to MM a shiny Pancham (which I never ended up getting, but I digress). I'd just beaten Grant, and was passing through Connecting Cave when I encountered a shiny Axew. Once again- I was stunned for about ten seconds, before I started muttering to myself, 'Holy crud, that's a shiny Axew!' Wasn't thinking straight for most of the battle. Didn't really regain my senses until I caught it, nicknamed it 'Tuscany', and saved. Then I was honestly overjoyed for the rest of the day (and even happier when I found a shiny Mime Jr the very next day).