Welcome to the adventure.
I would suggest you to continue the game as it progress.

Training mini-game can be abit complex for normal pokemon player as it involve some stat/EV stuff,
but you can take it as it's a mini-game that can make your pokemon stat goes higher.

Pokemon Amie (petting) can raise your pokemon affection toward you.
When pokemon affection level is high enough you may obtain bonuses in battle (higher critical chance, self-recovery, etc).
But bonus only applies in offline (single-player) mode though.