In Final Fantasy 1, beat the game and go to the title screen, choose Continue and press A on the beaten game (with the star next to it) it will start a new game, but once you name your characters and start save it, it to will have a star next to it, you have entered hard mode.
The enemies are harder here, Ogres and Gigas Worms will appear on Cornelias continent already, go to Cornelia and buy new magic and weapons and start leveling up.