I was still a young-un when I got the Mega Drive. I remember my dad buying me a Master System first, and then trading it in within a month or something for a Mega Drive.
I loved the Master System and when we went to exchange it, I was really upset because I'd come to enjoy it so much. My dad kept telling me 'trust me, the Mega Drive is the better one!', but I didn't believe him. It wasn't until we got it home and played Sonic 1, Alex Kidd (or Alex the Kid as I called him) and Altered Beast that I was convinced.
Sonic was the game that defined my childhood video-gaming. My favourite memory being staying up late with my dad as we played through the whole game. It probably wasn't that late, but to me it felt like I was staying up all night. My dad playing and me sitting on the floor watching while my mum watched from the couch. When he finally finished the game we all cheered.
I also remember the hype we all had for Sonic 2, my dad showing me a preview in a magazine (I think it was Sega Megazone, a magazine i still have some issues of, but I've been trying to find scans/archives of it for a long ass time). We did a similar all day Sonic 2 session where my dad finished that game and declared himself the 'Sonic Master'.
If we are talking about 'Genesis Era' and not just the console itself, we used to go to the local arcade near my grandmothers place, and my dad would play Afterburner with me all the time. They had the full rotating and tilting cabinet, it was unreal. A few years ago when Afterburner Climax came out, I was playing it and told me dad
"Hey look, it's a new Afterburner! Remember this game?"
"Oh wow. Hey! Does it still have the same music?" and then he started humming the Afterburner theme (this is like, 20 years after the fact).
My dad is awesome.