I tend to get more into even generations than odd ones, and I miss the new phases of several of those games. The two big ones for me are probably Gold and Diamond.
When Gold first came out, I was the first one in my group of friends to get it, and unlike with Red (which I got late), I had no one to spoil the game for me. I loved the sense of exploration and discovery, and it's something I've lost in newer games. I also remember the joy of realizing that I could return to Kanto and tackle eight more gyms in all their nostalgic glory. And beating Red for the first time was awesome (even if it was a bit of a letdown, because I expected to struggle, but didn't).
With Diamond, I have a lot of memories of online gaming and trading, many of them on NSider's forums before they shut down. I remember breeding highly desirable Pokemon to use as trade fodder, trying to grab shinies in giveaways off the GTS, and lots of other excitement, mostly during the summer after Diamond released. As someone who had never had the opportunity to go to an event and collect event Pokemon before, it was also great to finally start on my event collection via online trading.