Aaaah I'm fairly new to the gaming world for actually really playing myself, but one of the first games I played as a kid was Duke Nukem 3D. I would watch my dad playing it (yeah... I was probably far too young to be even watching him play it...) but I got interested and wanted to try it myself. I sucked at it (naturally) and lost interest for a while and mostly stuck to casual games for a long time (RollerCoaster Tycoon, The Sims, Pokemon, Crash Bandicoot Crash Team Racing... etc.)
However, what really got me into gaming was playing Assassin's Creed. I started playing the series a little bit after Assassin's Creed: Brotherhood came out, but I started from the original game and moved on from there. I was hooked into it immediately and I never looked back. Skyrim was one of my other first big-time games that really cemented my love for video games. I remember watching the E3 trailer for it and thinking it looked like the most amazing game ever (having never played a Bethesda title before). I got it the Christmas of 2011 and never turned back. Skyrim's now my favorite video game of all time, but now my collection of video games has grown to nearly 200 on steam.
Over the years, I've had an N64, a Playstation, a Playstation 2, a Wii, and Xbox 360, and soon I will have a full gaming PC (though I have been running Skyrim with a shitton of performance mods on my 3 year old laptop with integrated graphics... *shudder*).