I have lots of amusing memories of the first generation. I was the same back then as I am now, so I enjoyed performing bizarre stunts and using Pokémon and strategies that no one else ever considered using. ^_^
I obtained both Red and Blue around the same time, and I used lots of unevolved Pokémon to whom no one else paid any attention. I dominated in link battles, however, because I was pretty much the only person to use strategy in battles, and my strategies were very weird and unpredictable and compensated for the generally lower stats of my unevolved Pokémon. For instance, I had a Substitute/Leech Seed Bulbasaur two generations before anyone discovered how useful the combo was (at least, I never heard about SubSeeding until Ruby and Sapphire came out). Plus, it was just plain entertaining to one-hit KO a Dragonite with an Ice Beam from a Jigglypuff. XD
In my Yellow Version, I decided to see if I could defeat all of Brock's Pokémon using only Pikachu...and we actually did it. My Pikachu used Double Team and Growl twice apiece and then proceeded to beat down Geodude with a lot of Quick Attacks. Pikachu then used Growl twice on Onix (and used it again whenever Onix used Screech) and knocked out Onix after even more Quick Attacks, very nearly running out of PP for Quick Attack in the process. I only needed to use one Potion (which was fair, since, if I recall correctly, Brock used one, too).
Also, I read in a player's guide (Prima, I believe) that "no Pikachu, no matter how well trained, can possibly stand up to Lt. Surge's Raichu, so don't try to imitate the show"; being the slightly crazy person I am, however, I decided to prove the player's guide wrong and use my Lv. 28 Pikachu to defeat Lt. Surge's Lv. 27 Raichu. Surprisingly, Pikachu pulled off an incredibly easy win. If I recall correctly, it only took two Body Slams to KO Raichu (one may have criticaled, but I can't remember :/) and Pikachu never even took damage. Apparently, my Pikachu has unusually high stats for its species, because it has consistently pulled off improbable wins such as these. Who says Pikachu isn't powerful? ;P
I remember, though, how irritated I was that some Pokémon could only be obtained in in-game trades. I really wanted to use a Farfetch'd but didn't want it to be called Dux. Needless to say, I was very happy when I could catch wild ones in Yellow. ^_^
On a side note, remember those irritating rumors that were spread back in the day? I always thought the whole "Pikablu" shenanigan was ridiculous, but some of the supposed methods to obtain it and other nonexistent Pokémon were pretty hilarious. I remember rumors about searching for messages in bottles, using the Itemfinder 100 times in Mt. Moon, and sneaking into "Bill's Secret Garden." Ah, good times. XD