Hmm, I guess there's 2 sorta moments.
The first game I ever played a little bit of with my brothers would be Sonic Chaos on the Master System.
Pretty much all the games I first played was with them, taking turns and the like. We weren't exactly well off and we only had the Master System, a chipped PS1, and the PC around the time when the PS2 was out so there was a lot of fighting to play. Being the youngest, I got the least game time. We borrowed an N64 from some neighbours and I watched my brother play Ocarina of Time all the way through, loved it.
The first game I ever sat down and played myself, without any of my brothers, I remember to be Final Fantasy 8, on the PC. When my brother clearly stole a PC or something, though he's always denied it, and gave it to me for my birthday, I finally had time to play on my own. And it was really the game that made me think "Wow, this is so much fun!" And then I went onto games like Half Life and Deus Ex and it really made me realise games are spectacular things, and I loved the interactivity over just watching like in a movie. Only could afford a new game every few months so made use of many mods.
^ I've always said growing up like that is also the reason why I complain less about stuff in the Industry. I think people take it for granted how lucky they are to be able to enjoy a quite luxurious hobby like gaming.