Some game mechanics:
Player levels - The player levels up separately, and can gain XP by doing tasks.
Days - The game has its own day/night cycle, where you can sleep during in-game nighttime to skip to the next day.
Battling - Unlocked at level 10, and is self-explanatory.
Rocket Raids - Team Rocket will try to steal your Pokémon every few days.
Raid Battles - A team of Trainers gather up and fight a powerful Pokémon!
Events - Periodic things that I can't fit in the character limit.