One wall of text, coming right up!
My earliest gaming memories come from playing at a friend's house. I was about four or five, and my friend was a teenager from church. My parents and his parents were very close, and so my family visited their house almost every Sunday. As soon as I had enough spatial reasoning to hold a controller and control a character on the screen, my friend thrust me into the world of video games. He had a SNES, a 64, and a bunch of games for each. While I don't remember too much of my time with him, I do have fond memories of Kirby Super Star and Mario Kart 64.
My family moved shortly thereafter, and I made new friends closer to my age. One of them invited me to his house and showed me his Sega Genesis. The first game he showed me was a title I remember little about - it was some kind of puzzle platformer with ladders and a castle setting. He only played one level before switching to the main event: Sonic the Hedgehog 2. I fell in love immediately. We took turns playing, and as time passed, we figured out that the second controller moves Tails (I think I made a correlation from Kirby Super Star's partner system here). I still remember the day when we made it to Hill Top Zone for the first time; we were so excited that we lost our last life within seconds, and we couldn't replicate the feat for days.
Through all of this time, I didn't have a console or any games of my own. That changed on Christmas Day of 2000 when my parents got me a Sega Dreamcast with Sonic Adventure and Namco Museum. Of course the first game I tore into was Sonic Adventure. I wanted to read the instructions first, but my parents wanted to watch me play, so I consented and jumped in with my only prior knowledge being Sonic 2, a 2D game. So I tried to fight Chaos Zero (the opening scene), moving left and right only, jumping without knowing how to homing attack, trying to land on this thing's head (which I miraculously did once). My parents were confused too, and so we all tried to figure out what was going on... As I tried to wrench the controller out of my dad's hand, I accidentally hit up on the control stick. Sonic moved forward. Suddenly everything made sense.
Thus was born my greatest memories of my early gaming years. But I didn't play by myself.
Coincidentally, one of my parents' friends sent me a CD pouch that Christmas. As I went to store Sonic Adventure in it so I could try out my other game, I noticed that the front pocket had five bucks in it. These pockets were see-through, so I noticed $5 on the other side too, and turned the page. Sure enough, the next pocket had $5... and the next one... and the next one... In fact, every pocket had $5 in it. Of course, I spent this money on more games over time, among them Sonic Adventure 2, Chu Chu Rocket, Daytona USA 2001, and Rayman 2: The Great Escape.
I played my Dreamcast so obsessively that my parents actually put limits on how much I could play. I couldn't play during the week - that was when my homeschooling took place - but if I performed well enough, I could earn an hour on Friday. On Saturday I could play from when I woke up until noon (which prompted me to wake up at o'dark thirty) and an extra hour from four to five. I invited my Sonic 2 friend (my best friend at the time) over to play during this hour one day, and we battled each other on Sonic Adventure 2. It was so much fun that I invited him the next Saturday. The Saturday after that, he was unable to come, so I brought another one of my friends instead. Both of them came following that.
I began inviting more and more people until this became a regular thing - Saturday at four o'clock was Sega Day every week. My dad served popcorn and Kool-Aid as we took turns playing each other at whatever game we wanted to play that week (mostly SA2). It became a major part of our lives; we talked about how someone or another would win next week, or waited to redeem ourselves in a grudge match. One week we held a Daytona tournament, which ended up with me and my best friend in the finale with my friend barely edging me out on the last lap. During a major storm on a Wednesday, my dad assembled my friends out of the blue and we held an impromptu Sega Day until the bulk of the storm had passed. We kept this up for about three years.
But yeah, that's my early gaming years. Good times.