As of v1.1 this addon is both a data and resource pack so you need to install it as both. It will still work without the datapack installed but you won't be able to obtain the Partner Pikachu.

Singleplayer:

Put the .zip in /saves/[WORLDNAME]/datapacks
Put the .zip in /resourcepacks
Enable in game ESC - Options - Resource Packs

Servers:

Put the .zip in /world/datapacks
Sometimes it's called myworld, or something similar, sometimes it has a name instead
If you're in doubt about the correct folder, consult your server providers guide on how to add datapacks

Every Player has to have the .zip installed as a Resource Pack
Best option if for them to have it installed and enabled in their clients
It's possible to add them to server resources, but note that there have been issues with this for some cobblemon addons, so try whatever method suits you better

"Based on the appearance of the protagonists in the Let's Go! series games, along with the appearance of Starter Cap Pikachu in certain Pokémon games."

What's different about it?

1. Has a 430 Base Stat Total (45 HP, 80 ATK, 50 DEF, 75 SPA, 60 SPD, 120 SPE) and 31 in all IVs
2. Gets access to all its exclusive moves (Zippy Zap, Floaty Fall, Splishy Splash, Pika Papow)
3. It's a bit smaller than regular Pikachu and can't evolve nor change its cap
4. It changes from the medium fast to the medium slow experience group

Obtainable by having your Pikachu use Volt Tackle 10x and have 160+ Friendship
Note that Pikachu only learn Volt Tackle via TMs or Tutors so you need some extra mods!

"Based on the appearance of the alternative continuity Pokémon League Expo hat, Ash Ketchum's first hat from the alternative movie continuity of I Choose You!, along with the appearance of Partner Cap Pikachu in certain Pokémon games."

Obtainable by giving Pikachu Pewter Crunchies to hold (in its cosmetic slot)

"Based on the appearance of the Pokémon League Expo hat, Ash Ketchum's first and original hat from the original Pokémon anime series, along with the appearance of Original Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu a Rage Candy Bar to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's second hat from the Ruby and Sapphire Pokémon anime series, along with the appearance of Hoenn Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu a Lava Cookie to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's third hat from the Diamond and Pearl Pokémon anime series, along with the appearance of Sinnoh Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu an Old Gateau to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's fourth hat from the Black and White Pokémon anime series, along with the appearance of Unova Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu a Casteliacone to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's fifth hat from the XY Pokémon anime series, along with the appearance of Kalos Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu a Lumiose Galette to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's sixth hat from the Sun and Moon Pokémon anime series, along with the appearance of Alola Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu a Big Malasada to hold (in its cosmetic slot)

"Based on the appearance of Ash Ketchum's seventh and final hat from the Sun and Moon Pokémon anime series, along with the appearance of World Cap Pikachu in certain Pokémon games."

Obtainable by giving Pichu, Pikachu or Raichu Smoked-Tail Curry to hold (in its cosmetic slot)