Since your character recently moved to The Alola Region, instead of receiving a Trainer Card, you'll be given a Trainer Passport. You take a photo of yourself as you begin your game, but you get to take a new one every time you customize your trainer. (A player can retake their Passport Photo at any time by going into their Passport.) This Passport acts just like a Trainer Card, showing your Name, Trainer ID Number, Money, Battle Points, etc. Since there aren't any Gym Badges to show off this time around, you actually earn Stamps that are shown through your Passport. Each Passport Stamp is unlocked by a specific method.
Passport Stamps
Official Pokémon Trainer: Beginning of Game
Melemele Trial Completion: Defeat Kahuna Hala
Akala Trial Completion: Defeat Kahuna Olivia
Ula’ula Trial Completion: Defeat Kahuna Nanu
Poni Trial Completion: Defeat Kahuna Hapu
Island Challenge Completion: Complete Island Challenge
Melemele Pokédex Completion: Register All Pokémon On Melemele Island
Akala Pokédex Completion: Register All Pokémon On Akala Island
Ula’ula Pokédex Completion: Register All Pokémon On Ula'ula Island
Poni Pokédex Completion: Register All Pokémon On Poni Island
Alola Pokédex Completion: Register All Pokemon in Alola Pokedex
50 Consecutive Single Battle Wins: 50 Win Streak In Single Battle Mode At Battle Tree
50 Consecutive Double Battle Wins: 50 Win Streak In Double Battle Mode At Battle Tree
Poké Finder Pro: Get Final Version of Poke Finder
50 Consecutive Multi Battle Wins: 50 Win Streak In Multi Battle Mode At Battle Tree