In Final Fantasy II, directly South of Kashuan Keep is a forest, and within the forest is a small patch of grass. Walk over the grass to enter the hidden Chocobo's Forest, a large field of grass. If you find a Chocobo in it, you can ride it.
Defeat the four fiends to unlock each new dungeon. Each features random maps, exclusive items, and bosses from other Final Fantasy Games. I recommend beating all four fiends before venturing into the dungeons.
After you've completed the main story, an additional option will appear at the Final Fantasy II main menu. Select Soul of Rebirth to begin a new journey...
In FF II there is a match-the-faces tile game that gives items and cash prizes. When you are on the ice sled, hold A and keep pressing B to bring up the game.
In FF2, if you level up the Toad spell to lv16, the faces on the memory game change to toads. Also, the prizes will change, yielding Genji equipment for many rankings, and a Masamune for a no-miss game.
In the whisperwind cove, on the floor where you must find a certain boy, there is one boy that says he wants a "Trance Kuja" doll.
Beat Final Fantasy 1 and 2 once to unlock a sound test.