Like in Pokemon Ruby and Sapphire, selecting your name on the start menu brings up your Trainer Card, a unique identifier listing your name, trainer ID number, and achievements in the game. As well, your badges and play time are recorded. On the back, these records are kept:

Hall of Fame Debut
Link Battles
Pokemon Trades
Union Trades and Battles
Battle Points Won
Berry Crush

Once more you are given special Trainer Stars after completely special tasks, which also changes the colour of your card. Do note the card changes with the number of stars you've collected, but not necessarily which ones.

The accomplishments necessary to get a new star are as follows:
Defeat the Pokemon League Champion for the first time.
Jump 200 times in Pokemon Jump and collect 200 berries in Dodrio Berry Picking
Complete the Kanto Pokedex, except for Mew.
Complete the National Pokedex, except for all Mythical Pokemon.

Finally, you can have a print taken of your Pokemon party for $50 at the Rocket Game Corner in Celadon City. Your options for print colours are: Normal, Black, Pink, and Sepia.