Your partner pokemon will help you through various things and has their own signature move to help you. The higher the level it is, after using it's signature move, the more chances you'll have for capturing the pokemon. This will also help you when it will evolve. The pokemon can learn moves but learns three at a time instead.
For pokemon who has either Normal, Fighting, Flying, Poison, Ground, Rock, Bug, Ghost or Steel as their Primary type will level up as follows:
Level 1 pokemon: 2 exp. to level up.
Level 2 pokemon: 4 exp. to level up.
Level 3 pokemon: 6 exp. to level up.
Level 4 pokemon: 8 exp. to level up.
Level 5 pokemon: 10 exp. to level up.
Level 6 pokemon: 12 exp. to level up.
Level 7 pokemon: 14 exp to level up.
Level 8 pokemon: 16 exp. to level up.
Level 9 pokemon: 18 exp. to level up.
Level 10 pokemon: 20 exp. to level up.
Basically, it's 2X"Level of pokemon"=exp. to level up.
For pokemon who has either fire, water, grass, electric, psychic, ice, dragon or dark as their Primary type will level up as follows:
Level 1 pokemon: 4 exp. to level up.
Level 2 pokemon: 6 exp. to level up.
Level 3 pokemon: 8 exp. to level up.
Level 4 pokemon: 10 exp. to level up.
Level 5 pokemon: 12 exp. to level up.
Level 6 pokemon: 14 exp. to level up.
Level 7 pokemon: 16 exp to level up.
Level 8 pokemon: 18 exp. to level up.
Level 9 pokemon: 20 exp. to level up.
Level 10 pokemon: 22 exp. to level up.
Basically, it's the same thing as the leveling up system above me but +2
(( 2X"Level of Pokemon"+2=exp. to level up. ))
If you have any questions or suggestions, post it down here. Oh and, how do they gain exp.? By every time it uses a move on a pokemon that you will capture, once you capture it, it will gain one exp.. Also, if you ever decide to do a pokemon battle if you ever get some free time, if your partner pokemon wins it will get the same amount of exp. the opponent's level was. Please note if your ever going to battle another pokemon you can't use the pokemon you captured with your capture styler.