I don't know about my first memory, but one of my earliest is a pretty good story.
This was years back, on the Xbox 360 Edition. We tended to just play on the pre-made tutorial worlds (these were rebuilt every once in a while), so we didn't do much of our own base-building. After doing a few of the normal new-player-stuff, breaking blocks, running around, trying to build a portal to the aether, I found and tamed a Wolf. I loved Wolfy (my very creative name for him), and eventually he got killed by a spider. This made me develop a fierce hatred of spiders.
So, now to the interesting part:
For some reason, early on in the game, monsters would spawn around you, even while you were asleep. Almost every single time I went to sleep, I'd suddenly wake up with a zombie attacking me. One time a spider dropped through the skylight I had made. Obviously, all this was very annoying, so it made this next experience feel much better. I wake up one morning, and hear a spider on my roof. I decide to execute this infernal beast for its impudence, and start building a staircase to the roof. The spider doesn't attack me (for some reason) until I'm standing on the roof for it.
For some context, I was living in one of the pre-built buildings, one with a giant, open chimney on the top. I start my fight with the spider, him leaping at me and dodging the swings from my (probably) wooden sword. Finally, I darted to the other side of the chimney. He took his last leap at me from across the flaming pit of hell that awaited him, and with one swing I avenged my fallen Wolf, and my enemy fell into the chimney.
Overall, very fun experience. I still gladly remember the sounds of my enemy's dying shrieks as the flames took him.