My favorite part has to be the collecting part itself. I’ve been collecting things all my life (nice-looking peddles and post stamps from the age of 5 or 6; to Pokémon TCG, Yu-Gi-Oh TCG, twisty puzzles, euro coins, minerals, gemstones, fossils, pogs, loads of virtual items (like TF2 weapons, Smite or Overwatch cosmetics), etc. etc. later in life.
What I really like about collecting my Pikachu, Mimikyu, Seviper, amd Moltres #21 WotC Artwork Variation collections however is that there is a checklist and a clear goal in mind. Finding a new card from that list, especially now that I’m nearing the end for the collections is a great feeling. Buying that card, waiting impatiently for it to arrive, and finally having it arrive are all great feelings of that collecting part.
Apart from that I also really agree @mjisaacs. The research surrounding collecting Pokémon TCG, and both sharing with and learning this information from this awesome community, is also a great aspect I deeply enjoy. I’ve already made a few articles, like my Base Pikachu Artwork variations article, 1st edition stamp variations article, or Teach Sets work-in-process information to name a few, which were all well-received and I enjoyed putting together and updating when new variations are discovered.
So those are my two favorite parts. Having said that, I enjoy and agree with almost everything named thus far. Admiring and re-organizing my collections; winning an auction at the last second; opening packs/boxes and hoping for the pulls you want; the artworks of the cards; the stories from certain cards or people here in the community; the nostalgia from my childhood; the memories from then, now, and in the future; etc. etc. All things I agree with, and all things contributing me to keep collecting Pokémon cards for a long, long time in the future. 
Greetz,
Quuador