Mine basically evolved with whatever I could get in some manner other than theft. A lot of it involves older relatives having stuff I could see but not touch.
First game memories are of the Xmas my stepmother got my father a 2600. I wasn't allowed to touch it but man was that thing sweet. It went into a closet shortly thereafter and I never saw it again. This is probably 1979-80.
At some point not too long after, my aunt got an Odyssey 2, but that was in her bedroom, so was off limits to me. Then I saw my stepmother's parents also got one, but also in their bedroom, and also off limits. Guess is 1981 or so. A couple years later same aunt is dating this guy who decides to buy several brand new cabs for her and put them in a spare room. I remember Pac Man and an Asteroids cocktail and there were about 3 more cabs. This was early to mid 80s... again, when we visited, not allowed to play them because my mother wanted me to go outside. Same house also had the whole lineup of Coleco tabletops, sometimes I got to play those.
At some point when 2600 was totally clearing out, maybe 85, I get a 2600 and some games to play on my 13" b/w TV. Man that was heaven. Bowling, Pigs in Space, Star Raiders was brilliant. My little brother and I used to play Activision Boxing and I would corner him and just pound his nose until he threw a snot tossing tantrum on the floor.
Christmas, probably 1988, my mother tells me I can get an NES if I can come up with half. I manage to convince neighbors that they need a lot of odd jobs done and come up with the $45 so I do get one. SMB all day and night. I tell my stepmother all I want for Xmas is an NES game and she gets me DK Jr, man was that a huge disappointment. All those awesome new games and she gets me one I had for the 2600. At some point in the next year my mother asks me if she can throw the 2600 away since I have the NES now. For some reason I tell her yes.
From then on, through middle school and high school, it's all watching Genesis at my friend's house, but not playing much as he was always into RPGs. We would also have huge Baseball Stars (NES) tournaments until no one could come up with an NES that wouldn't blink so much it erased the cart memory all the time.
I was out of games until Sophomore year in college my roommate's gf's little brother gave me an old TG16... I found the TurboList, found a clearance Duo, and was hooked. I was all TG16 until release day for the Saturn when I dropped way too much on a Saturn and the 3 free games it came with... played that a long time, then got an N64... played that a lot, sold it while it was still worth a bit in about 96 or 97. Was out of games again until I got my son an N64 for Easter in 2002... that was it, I got into console collecting bigtime for a couple of years. In 2003 I picked up a POW cabinet at a yard sale, that was the beginning of the end of consoles for me and the start of an even worse addiction, cabs. In 2004 I pick up a nonworking pin at a yard sale... pins are 10x worse than cabs.
So, here I am now, thinning out the console collection, refining the cab/pin collection... though still in the "most don't work" stage.
Man that post was longer than intended.