Episode GuidePlease note that this is not a complete episode guide (obviously). Eevee has appeared in tons of episodes as a minor gust or just making a cameo, but listed below are just a couple of recommendations if you want to see Eevee in action. These are some of my personal favorites. The descriptions a pretty brief, as to not spoil anything for anyone who wants to watch.
EP040 - The Battling Eevee Brothers: This is an old episode, and the first episode centered around Eevee. This episode revolves around a young trainer named Mikey, and his conflicted feelings about evolving his beloved Eevee. It's a very good episode, and one that I probably recommend above all others.
EP116 - The Rivalry Revival!: This is the first episode where we see Gary Oaks Eevee. It features a great battle between Eevee and Pikachu. It is also an older episode, but very fun to watch.
EP183 - Trouble's Brewing: This is a very Eevee-centric episode, focusing on Sakura's Eevee as well as the Kimono girls and their eeveelutions. All of the other eeveelutions also get a chance to shine, so this episode is definitely worth a watch.
AG157 - Time Warp Heals All Wounds: It doesn't happen until the end of the episode, but May's Eevee finally hatches from an egg. It's a very sweet moment, with even Meowth being touched and giving his blessings to the newborn Eevee.
DP174 - Last Call -- First Round!: Two Eevee's are used in the grand festival competition in this episode. They both evolve during the battle. This is a good episode to see Eevee in action.
BW102 - Team Eevee and the Pokemon Rescue Squad!: This is the introductory episode for Virgil and his Eevee rescue squad. Not only do we get to see tons of Eevee, but we see every evolution (minus Sylveon) as well. We also get to see Team Eevee in action with a rescue mission. While I always recommend watching the Battling Eevee Brothers (mostly because of the conflict with evolution), this episode is probably my favorite of all and the most enjoyable. A must watch for all Eevee fans!
XY 088, 089, 090: All three of these episodes are an absolutely must watch for Eevee fans. 088 centers around Serena and her quest to capture the super shy, but talented Eevee. It has some really cute moments with Eevee dancing, and ends with Eevee joining Serena's Team. 089 focuses around Serena and her friends trying to help Eevee overcome its shyness through various methods. It includes Eevee's first official battle aside from Team Rocket. Eevee spends the entire episode out of the Pokeball, so there are many cute moments. 090 is a continuation of 089, where Serena competes with her Braixen and Pancham. Eevee gets spooked by some fireworks and flashes, and the group has to find it before Serena performs. Eevee is extraordinarily cute at the end when it watches Serena compete with her Pokemon. All three of these episodes sort of go together, and are heavily Eevee based.
SpecialsPK25 - Eevee & Friends: This was a short that played before the movie Genesect and the Legend Awakened. It is completely focused on Eevee and all its evolutions, including Sylveon. The entire short is focused around them in sort of a story-telling mode, with each Eeveelution getting the spotlight. This is the ultimate treat for any Eevee fan.
Mewtwo - Prologue to Awakening: This was also a build up to the move, Genesect and the Legend Awakened. In this special, we get to see Virgil and Team Eevee in action. Not just doing rescue missions, but also working to extinguish a forest fire. Not all of Team Eevee can be present in this episode, but we get to see a good amount of all the ones who are present.
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