My earliest gaming memories come from a trio of games for computers, before it was even remotely fashionable for computers to play games (mid 80's). One was castle - an graphic adventure game of primitive quality. The objective was to simply escape a castle. The graphics were given courtesy of the ASCI special character set. I know not how much progress I made in the game but as far as I was concerned it was damn near unbeatable.
The second was alley cat, a slightly more advanced game. I cannot recall what the objective was though I do know the game involved avoiding junk being thrown out of windows while climbing up a series of obstacles.
The third was bouncing babies, and I have seen this concept reiterated many times over. Basically, you controlled a pair of firemen holding a trampoline. Babies were being tossed out of a burning building. Your goal was to use the trampoline to bounce the children to the safety of the waiting ambulance. As the game progressed you were forced to juggle many children at once.
None of those games really had much of an impression on me though. When the NES first came out and my parents picked one up, the only games we had were Super Mario Brothers/Duck Hunt (which came bundled with the console) and Golf (my dad's pick). For quite awhile these games failed to draw me into the fold.
The first game that I truly loved, and what probably represents my first step into geekdowm was Captian Skyhawk, which I got for christmas in 1990. As luck would have it, there was a war on at the time and the news was full of coverage of Allied planes bombing some target or another. I don't recall ever really connecting the dots in a specific way, but I can only assume the Gulf War probably had more than a little bit to do with my love of that game.
Other memorable milestones include the first game I played on the internet (Duke Nukem 3d, thanks to the third party program internet commit), the first game I purchased myself with money from something resembling employment (Unreal), my first MMO (Arctic, a MUD), and the first online game I actually became good at (Team Fortress/TF Mega for Quake I).