This isn't a very old game so I'd imagine this should be fresh in your mind.

My first memory was when I was at a friends house, he had Skyrim on the 360. So he loaded his save up and he was in the northern part of the map. (It was snowing) I remember running around with a claymore one hitting trolls left and right! XD

I had been a strong Oblivion fan and Morrowind enthusiast.
So this was a must buy.

I didn't see it anywhere (not counting those spammy screenshots of a troll and riverwood that were all over the net), I got the game right away. So the first memory is driving with Ulfric and pals hahah. Though some of the dearest memories are from when it was Christmas, girlfriend and I were tucked in a big bed, she was watching Christmas movies (and partly me playing cause she was interested as well) and I was running through the western snowy forests of the Whiterun hold, killing spiders and stuff. It was beyond awesome!

My first memory was a few days after release, I remember the Troll near Ivarstead by the river attacking me and since I always play games on the maximum difficulty from the beginning he practically 1 hit me and after that I pretty much always avoided that troll, even though I know he's the weakest variant of trolls now. I also play the game DiD so this was a very traumatic experience.

My first memory, is looking at Helgen Keep's stone textures. They were looking good at a distance, even back then -before the high res dlc and other texture replacements.

My first memory was being frightened during the execution when that Dragon landed on the tower and looked directly at me. Bethesda did a really fantastic job of modelling, texturring and animating Dragons in this game, in my opinion. It's only the second time I've been frightened in an Elder Scrolls game (the other would be my first encounter with a Greater Bonewalker in Morrowind).

A Skyrim Mod for Oblivion was dropped because of Skyrim being announced as the next Province to be featured in a Game.

Finding the first of many, many Stray Dogs, and finding out how Ancient Nordic looked on a Female character before meeting Aela the Huntress. Angis' Shack. A consistent CTD when entering an Ice Cave.

Hevnoraak getting forced into a corner by the Stray Dog, and becoming My first Dragon Priest kill in Skyrim.

First Named Weapon, Trollsbane. I had Ebony Weapons and a Shield, but this Warhammer had a name!

Finding a Hagraven for the first time, and discovering Fireball

Reload, discover a Briarheart and Two Hagravens.

Frightened? That's funny. I was relieved, thinking thank the gods that dragon showed up; I hope he burns this whole accursed place down. Alduin's my pal.

My first memory goes back to when one of the trailers was released and Forsworn were shown for the 1st time. I generally don't watch trailers but I watched this particular one, and I remember seeing their masks which I found great. Funny though, none of my characters yet ever wore those masks.