I really enjoyed choosing and training my starter (Squirtle, eventually a level 100 Blastoise). I also have fond memories of beating the Elite Four for the first time, and of catching Mewtwo (with my trusty master ball) after breakfast one Saturday morning with a friend. Years later, I recall completing my Pokedex, which was also immensely satisfying (I had completed my Gold Pokedex by that time, but then Gold died, along with a bunch of my Red Pokemon, and it was difficult to finish my Pokedex with the limited repertoire of games that I had at that point).
A few moments stand out as being challenging when I first played, and were thus satisfying when I finished them. On my first playthrough, I struggled to make my way through Mt. Moon (I even restarted my game to backtrack to the route I had noticed to the left of Viridian (i.e., the path to Victory Road), thinking that was the way to proceed). I also got stuck in Seafoam Islands at one point, and finally found the exit before successfully extinguishing my PP on all of my party members and blacking out. Victory Road was also very challenging for me when I first played, and I was proud of myself when I figured out where to drop the boulders and got through it.
In retrospect, I'm not sure why those elements of the game were difficult, but I guess I've grown a lot in 16-17 years. It still surprises me that I managed to lose against the trainer standing by the ladder I needed in Mt. Moon, and didn't think to trek back there to see what he was blocking (but rather decided to restart the game). Regardless, the difficulty and novelty of my first Pokemon experience made Red memorable in ways that later games were not (even though Gold is still my favorite).