Like the main Pokémon games, you can raise your Pokémon simply by defeating other Pokémon in the battle system. However, Pokémon Quest also provides another alternative to doing that in order to speed up the process. This is accessed through the Training button when you edit the team
The first option is the ability to increase your Pokémon's level. To do that, you have to put the Pokémon you want to level up in the top slot and put up to 4 Pokémon in the bottom four slots. This will use those Pokémon as training fodder and cause them to disappear in order to give experience.
High Level Pokémon, Evolved Pokémon and Pokémon with many Power Charm slots give more experience, and Pokémon of the same evolution line of the Pokémon you're trying to raise give considerable experience.
For each level you increase, you gain 1 stat point in each stat. This adds to the base stats, IVs and bonus from how it's obtained
Changing your Pokémon's moves is another aspect that can be found in Training. To do this, you simply follow the same mechanics as gaining experience, using up to 4 different Pokémon as fodder, but this time the Pokémon increase the chances of success. The higher the chance of success, the more likely it is that the move change will be successful
The success is determined, like the levelling up, by the Pokémon being used to train against with High Levelled Pokémon and Pokémon of the same evolution line giving great chances. However, the more you use this on a Pokémon, the lower the rate goes.
Unfortunately, there is no way to control the move you obtain through this method.