Gym Skill

There are 8 Gym skills you can improve on. Leveling these up will increase flat stats for your pokemons, and every 10 levels you unlock a 'core'. They will add percentual stats according to the level of inlaid core. You can gain additional stats from 'refine'.

Gym Stats
Gym 1 - Santalune Gym: HP

Gym 2 - Cyllage Gym: P-Atk

Gym 3 - Shalour Gym: Sp-Atk

Gym 4 - Courmarine Gym: P-Def

Gym 5 - Lumiose: Sp-Def

Gym 6 - Laverre: SPD

Gym 7 - Anistar: Crit

Gym 8 - Snowbelle: Res

Core Slots
Cores can range from white (lowest) - green - blue(3.5%) - purple(5%) - to orange(7%) (highest).

Refine
You can 'refine' the stats in each Gym by rolling the stats of different Attributes. This will also consume Gym Score Points. As you reach the higher end of each attribute range (90?) you will unlock higher ranges. This will be shown by the attribute's color as well, changing from blue (lowest) - purple - to orange (highest). The according slot to the heightened attributed will take on this color as well.