The following modes can be accessed on the Title Screen under an additional "Extra" menu option.
Unlockable
How to Unlock
Collection
Beat the main game.
New Game+
Beat the main game.
Records
Beat the main game.
Survival
Beat the main game.
How to unlock Achilles
After you beat the game, play through and beat Survival mode (which consists of 117 battles). Upon beating Survival Mode, you unlock a screen that says "A New Party Member has been added". To use Achilles, go to Survival Mode, and choose preset party. He will be at the bottom.
New Game +
After you beat the game you can start a new game with some of your stats and items intact.