I played my dad's Atari 2600 for all of maybe two months or so. Don't really remember any of it except some kind of horizontal space shooter with aliens that were represented by these red blob sorts of things.
NES was the first system I owned, got it as a present when I was around 5. Had Mario, Mario 3, Zelda, Metroid, Startropics, and some other random stuff that was either forgettable, like Pinball, or stopped working too shortly after I bought it--Adventure Island, Bubble Bobble, Blaster Master. (I would completely buy Startropics, Bubble Bobble, and Blaster Master again on VC, though.)
Actually, it's probably ironic enough that some of my fondest "gaming" memories from really early childhood consist of
the Blaster Master book and the Wizards and Warriors book, which I can't find a picture of, but which was awesome because the two characters save a maiden in basically every chapter and eventually it starts feeling unintentionally ridiculous. And can't forget the Mario choose your own adventure books, either. Pretty cheesy, and there was one that makes me want Toad playable in SSBB, where they held a Mushroom Kingdom Olympics and Toad turned out to be more buff than the Hammer Bros or anyone on the Mario/Luigi/Toadstool team. >_> (They were probably drawing from SMB2, where he picked things up the fastest.)
...That was all a tangent. I still wouldn't say NES was the console I "grew up with," probably because I bought an SNES when I was 7 and generally didn't look back. (There were a couple exceptions, like Dragon Warrior IV.) It would probably be hard to overstate how much time I dumped into Super Mario World, then later Secret of Mana, Chrono Trigger, Final Fantasy VI, Yoshi's Island, DKC2, Earthbound, Illusion of Gaia, Kirby Super Star, etc. And I do think I remain heavily influenced by all these games. I could point out that most of these have silky smooth controls, but that's really true of any classic titles on any system and I would have had my standards of responsiveness set by something else if not by these games. So instead I'll point to the fact that every one of them has an awesome soundtrack. To this day I value play control and music most highly, so--perhaps that latter part can be attributed to these early impressions.