Once you have some Blank Cards, you can visit NPC Marie. Marie will sketch a Pokemon in your party on one of your Blank Cards. Once this is done, that specific Pokemon may never model for Marie again (though others of the same species will be able to). A Card with a Pokemon sketched onto it will teach any move the Pokemon knows for the first 20 Levels to any Pokemon that can learn it. To teach moves that the Pokemon learns at higher levels, you will need more cards of the same Pokemon. Each card with the same Pokemon equals twenty more levels.
NPC Regigigas will allow you to turn your Pokemon Cards into actual moves for your Pokemon. For example, if you were trying to teach Blaziken Flare Blitz (a move it learns at level 66), you would need 4 Blaziken Cards, or 3 Growlithe Cards (since Growlithe learns the move at 56, 3 Growlithe Cards allow you to teach any move it will learn up to level 60), or a combination with another Pokemon that learns Flare Blitz. Check before-hand to find out the optimal method for teaching your Pokemon moves.
NPC Registeel will allow you to enter into battles with opponent AI Pokemon. The battle system is still in the process of being developed, so opponent Pokemon will know four random moves, and usually those moves are moves that the Pokemon would not normally be able to learn. For beating these AI opponents, your Pokemon will gain experience and you will gain BP (Battle Points), which can be spent on prizes.
To find all of the NPC's mentioned in this article, check the first post here: http://forum.vdexproject.net/viewtopic.php?f=44&t=19758