I have couple that spring to mind.
Getting my first Level 100 pokemon in Ruby (I had never had a level 100 pokemon in my other games, had beaten the elite 4 and did the post story things, like beat Red in Silver, but never a level 100), I remember grinding Minun for weeks, and trading him to my completed Fire Red, and taking on the E4, then to collosseum to take on Mt. Battle, and beating my opponent at Battle 100, then Minun himself, levelled up to 100. I had a few Pokemon who were in their 90s, and remained in their 90s for about a year or so, then as DP was coming out, I started levelling them up to 100 ready for the release of DP.
My other achievement leads from Minun, that level 100 Minun has remained in my teams ever since, and it was a sort of special moment and proud moment for me, when in Alpha Sapphire you can get a certificate for transferring Pokemon from the original games over, so Minun being the first one to get a certificate felt special too me.
Another proud moment is not something I've achieved per se. But I've recently started a Pokemon Club in school, and introduced the games to a lot of the children, they only played the cards and watched the show before, but now a group of them have played the games through the club and its so nice to see them sharing advice, trading and battling against each other. And then sharing the knowledge that I've taught them, onto their younger siblings and also to the younger children in the club. Last week, I brought my Nintendo 64 in with Stadium 2, and was watching one of the youngest children battling against an Umbreon, then one of the older children who I've given tons of advice too, telling that child to use Fighting Moves on Umbreon because Fighting is Super Effective. Its not much, but it was a proud moment, a few months ago he had no idea about type advantages and disadvantages and now he's there telling others what types work well against others.