I play all genres except sports - exceptions would be arcade sports games. There have been relatively few of those (good ones, anyways) in recent years, so I've stopped playing entirely.
My favorite genres seem to have ended up being RPG and Strategy/Tactical games. If I had to narrow it down, I'd say my favorite game of all time is Chrono Trigger, with Fallout 2 in second, with Zelda: A Link to the Past/Ocarina of Time, Elder Scrolls III: Morrowind, Final Fantasy Tactics, Chrono Cross, Final Fantasy VI, Super Mario Brothers 3, and Ogre Battle: March of the Black Queen following those up. Also very fond of Guilty Gear and a multitude of other games, I'm sorry for not being able to narrow the list down further. I've greatly enjoyed the Mass Effect series, and have been desperately wishing to play the two Dragon Age titles. I seem to spend a lot of time playing Mount & Blade: Warband as well.
I'm sorry for not being able to narrow down the list, and for leaving many newer titles out that I adore (Such as Guilty Gear, mentioned briefly, KOTOR, Katamari, and so on). I suppose the entire Super Robot Taisen series deserves a mention for me being addicted enough to it to play them untranslated and figure out the game mechanics, despite not being nearly fluent enough in Japanese to do so without constant trial and error.
As to how I "got into" gaming, my brother owned an NES (It came out a little after I was born, so I was in the prime position to get into it upon reaching an age where I had sufficient coordination!). I received an SNES the year after it came out, with a copy of Zelda: A Link to the Past, followed later by Chrono Trigger and Ogre Battle, as well as Street Fighter and a plethora of other titles. I'd go on to buy a N64, PSX, XBOX, and 360 - I really ought to get a PS2 or 3 someday, if only because so many wonderful titles remain exist (And PS2 titles really are dirt cheap at this point!)