Ehh, hard to think of a specific moment right at the top of my head, but I generally loved the entire game, juuusstt apart from the fact that about 1/4th of the game took place on water >:[ I guess some of my favourite moments would probably be beating the E4 for the first time and finally capturing the Regis. Reason beating the E4 is special, is pretty much because it's one of the hardest E4's imo. I don't know if I've just been super lazy each time I played through the games when it comes to balancing my team and their levels, but I've generally found R/S/E to be some of the hardest games in the main games series, and with the Victory Road and the E4 just shining up there at the top. My Pokemon were almost always heavily underleveled and super weak, and the only reason I beat the E4 in my most recent playthrough, was because I did a cop-out and just used Kyogre instead of my Ludicolo. But in my very first playthrough, I spent so long trying to get through it. I honestly think I just gave up at one point and started over, but I don't quite remember. I just remember the happiness I felt when I finally claimed my victory against Steven and his gosh darn Metagross ;D And one of my more memorable moments actually came right after that, but I wouldn't really refer to it as a "good" moment.
Immediately after beating the E4, I went straight ahead to catching Rayquaza. On the way to the Sky Pillar however, I encountered Latios. Only problem was, I didn't actually know it was a legendary, so I just ran away. I don't know why it didn't even occur to me, that it wasn't a Pokemon I ever had seen before, and the fact that it looked legendary should've at least meant something to me. But no, I was too set on catching Rayquaza, I didn't even think about it :p
As for my other greatest moment in the game, it was probably finally figuring out the braille, and meeting the requirements to go through them, and finally catching the regis. I mean seriously Game Freak, nobody reads the game manuals, why would you include the decryption codes in it, especially if you're targeting the games at kids who are likely to throw those away :( Or was that just me? Either way, it wasn't until one of my friends found the braille codes in the manual (I'm pretty sure it was in there? It's been years) that I finally managed to get myself some Regis. Right after I got my mom to translate the English for me ;D