Oh boy, where to start...
I watched the anime before I played the games, so the first time I played Pokemon Red I thought my Charmander could win against Brock's Rock Pokemon through sheer willpower or exploiting a weakness. After limping my way out of the Pokemon Center for the tenth time, I broke down and caught a Nidoran to Double Kick them to death.
Actually, the very FIRST game I played was Pokemon Silver (bought a Gameboy Color for it and everything!), but I couldn't figure out how to switch the first Pokemon in my roster out with a different Pokemon, so I was frustrated that my starter would always come out first and mop up all or most of the experience points, leaving it over-leveled and my other Pokemon under-leveled. I played through the whole game with each starter before I thought to ask one of my classmates how to switch the first slot with a different Pokemon. I was so embarrassed when they showed me the easy way I'd overlooked.
When Gen 3 games came out, I thought Torchic would learn a steel claw move the way Charmander does in the FR/LG remakes, so I kept leveling up waiting for it to learn a Steel move to use against Roxanne. Before it evolved (where I would have discovered Combusken learns Double Kick), I broke down and caught a Lotad, then it got its under-leveled tail kicked time and time again until it evolved into a Lombre, when I FINALLY won my first Gym Badge at the 36 hour mark. I was so embarrassed by this mistake that for years after I would speed run through all my games, not wanting to take forever like I did on that first Pokemon Sapphire playthrough.
Anime-wise, before Johto Pokemon were officially revealed, I believed the rumor that the round mouse Pokemon (Merrill) was called "Pika-Blue," a new, alternate evolution to Pikachu.