Sky Pillar is a daily game mode that is unlocked at lvl 60 together with DNA Skills. In this game mode you complete several battles across the floors until you reach Floor 200, wich is the highest floor at the moment.
Gym Challenge is unlocked at lvl 50. In this game mode you can earn yourself Gym Points so that you can level up and refine Gym Skills to get Stats. These stats are applied to all of your Pokémon.
Legendary Journey is unlocked at lvl 55 together with the Wish Star Platforms.
Abilities Are like extra passives that your Pokémon can have. Each ability impacts the way that your Pokémon performs in battle.
Pokémon Transport is unlocked at lvl48. In this game mode, you can send multiple teams of 3 Pokémon to transport materials and receiving rewards in return.
Accessory or Gear is unlocked at level 45 and it's one of the most important things in your Pokémon. Having the right Gear Set might be the difference between a win or a loss. Equipping an Accessory to your Pokémon will increase his stats, and some even give you additional buffs when using 3 pieces of the same type of gear.