As is the nature of the genre, one of the main interests of the players will likely be to obtain as many different kinds of pokemon as possible (gotta catch ’em all). This brings up the question of how to handle actually keeping them all. In the show, Ash would only ever keep a small number of pokemon on him at any given point, and sent all of his extras to Professor Oak to take care of in his stead. This is similar to how I would like to handle it in the game.
Players will only be allowed to keep six pokemon on them at any given time (yes, just like in the video games). If they have exactly six pokemon with them when they capture a pokemon, the pokeball used to capture the pokemon will become inactive and unusable until the player gets to a computer and uses it to transfer their pokemon to someone to take care of for them until they need it back. The person who takes care of the pokemon for you can be anyone from your mother or a childhood friend back home, to a pokemon daycare center (and no, they will not level your pokemon for you…nice try).
Sending and retrieving pokemon is as easy as finding your local pokecenter. Pokecenters have computers that allow you to call back home via video phone and an item transfer system that instantly teleports items, including inhabited pokeballs, to and from any location that you are contacting via video phone. This also means that friends can store items for you as well, though I don’t foresee this becoming an issue.
Storing pokemon with your friends in this way does, however, mean that they will not gain any experience from fighting and therefore are easily left behind as far as power level goes. They also will not have a good relationship with you if you never use them. So, it becomes a decision of what kind of trainer you want to be. Do you want to raise all of your pokemon equally, thereby having many options to choose from; or do you simply want to raise a few of them exclusively, having more direct power but less versatility?