My most vivid memory is of the Safari Zone. It wasn't the prospect of catching rare Pokemon, though the notion did have appeal to me. No, what makes the Safari Zone stick out in my mind is the quest for HM03, Surf.
Any R/B/Y veteran will tell you the significance of this; unlike in later generations, where you receive the vital HM from a mentor or some such at a key point during the narrative, in Generation I, you've got to find it yourself, and without any clues to go on, this is not an easy thing to do. Luckily, I had a friend who was reasonably well-versed in Pokemon, and he was glad to tell me that HM03 could be found in the Safari Zone. He didn't say where, just that it could be found there.
Now, under normal circumstances, this wouldn't be a problem. The Safari Zone is a pretty big area, but by the time you get to Fuscia City, you're experienced enough in exploration to get around easily enough. The problem is the time limit. That horrible, horrible time limit. 500 steps, and 500 steps only.
I was stuck at this part for days. I'd pay my 500 zenny, meticulously comb the area, run out of time and start all over again. It seemed as though the HM was always just beyond my reach. If I could just have a few more steps, just a little more time, I would have had it.
The day I finally managed to find that secret house remains one of my greatest gaming triumphs. It might seem small and insignificant, but it was a huge victory for me, and as frustrating as it was at the time, I wouldn't trade the memory for anything.