Most of these are games only. I don't have many memories regarding other media.
-February 2009: Finding my first shiny - a Starly in Diamond. Unfortunately, this was also in the Great Marsh, so he was also the first shiny I lost.
-March 2009: Completing my first Pokémon game - Diamond. The memory's a bit fuzzy and tainted now, but I never really forgot how difficult Cynthia was to beat in that game. Thankfully, I had an easier time in Platinum.
-June 2009: Speaking of Platinum, the massive tantrum I threw when I thought I had gotten stuck in the Distortion World. Got out of that one just fine, and even caught my first shiny in that game - an adorable blue Psyduck - afterwards!
-Early 2010: Playing SoulSilver for the first time with an emulator and a translation patch upon release because I just couldn't wait to see what this game had in store. And on that note, scrambling to find a previous version of my save file after I'd caught Suicune because I couldn't believe Eusine was gone forever.
-June 2018: Thus, the massive relief I felt when I played Crystal to completion later on and found Eusine in Celadon City after catching Suicune, just as all those playthroughs said I would. He was there waiting for me to catch the rest of the beasts! 😭💖
-December 2017: My Totem-sized Mimikyu OHKO-ing Ultra Necrozma in US/UM after some setup. I think this is the moment that cemented Mimi as my favourite Pokémon. A scruffy little baby slaying the most powerful Pokémon at the time? Badass.
-November/December 2018: Playing LGPE for the first time in my Switch. It was the most fun I'd had with any Pokémon game - and still is, despite the Kanto-mons only Dex. Oh, and me simping losing it over Koga's redesign in this game on Twitter, as I was even more into him back then. (Mind you, I still think it's the best he's ever looked, to the point I replayed LGPE late last year just to see him again. 😏 Hope Johto re-remakes keep his design among similar lines).
-November 2019: Getting a Mimikyu within my third or fourth Surprise Trade in Shield. I was super miffed that Mimikyu was locked to post-game because I wanted one in my team. I was absolutely floored when she came in, and her son joined my team almost immediately after.
-March 2020: Completing my first Monotype run - Ghost-type, on Ultra Sun. (Also notable for being the last run of a Pokémon I made online updates for. They were hampering my enjoyment of the game itself, and I dropped out of most social media later that year anyway.)
-November 2021: Getting back into Pokémon after a long absence upon the release of BD/SP... except that instead of getting the new games, I replayed LGPE as I mentioned before. It was also liberating to admit to myself that Kanto and Johto were my favourite regions, as some people I used to be close with would have branded me a Genwunner/nostalgia junkie otherwise.
Just one more thing: OP, you might want to change the title to something like "favourite Pokémon-related memories" because a lot of us are sharing these now. Oops? 😆