The Gym is an area that allows you to level up creatures while you are offline or doing other things in the game.
Currently the only Gym available is Coachroach's within the 2nd area of Onyx City, which becomes available in Episode 3.
Gym Passes are needed to train a creature. The number of Gym Passes needed increases as more training days are chosen to encourage players to at least revisit once a day to save more coins and/or credit.
Gym Passes are purchased at the gym, and you are given the option to purchase them for a small amount of credit, or a lot of coins, whichever you happen to prefer or can afford.
As you go through your list of creatures, the menu will update and show you, approximately, how much EXP they will receive by the time the selected days of training finishes.
You cannot train a creature if any of the following is true:
Your creature will gain 1/24 of the experience you saw listed for a day's training each hour. However, a creature will not actually level up until it has returned to you.
When a creature's training is complete you will receive an in-game mail notification alerting you about it. Please keep in mind that once a creature's training time ends, it will no longer receive experience and is merely idling until you recover it.
A creature's training can be cancelled any time by returning to the Gym and recovering it. However, Gym Passes spent on the training will NOT be reimbursed to you should you choose to do this.