Level up White Mages:
In Astos' Castle, the northwest castle, is a room with three treasure chests with Undeads guarding them. The two upper ones are Mummies. Use two Harm spells to wipe them out to get 150 to 300 experience, and 600 to 1200 experience for using just two spells. This is most useful immediately after getting the key from the elf prince.

Strong Monk:
If you use a Monk in your party, do not equip him with any weapons. His attacks will be much stronger when using his own fists.

Adamant location:
The location of the Adamant needed for Excalibur is in the tower in the sky -- you need the chime and the cube. Go in the tower and continue to the place with eight ways to travel. Go in the bottom left diagonal and fight the enemy that guards it. Open the treasure chest to get the Adamant.

Original mode:
Successfully complete the game to unlock the "Original Mode" option, which allows the NES version of the game to be played.

Dragon Warrior reference:
Find the three tombs located next to the white magic shop in Elfland. The tombstone farthest to the left reads:

"Here lies Erdrick"
"873-866"
"R.I.P."

This is a direct reference to a character in the game Dragon Warrior.

Easy experience:
Northeast of the port town of Provaka is a two space wide peninsula with enemies from the Northern Continent. They offer a lot experience and are difficult to defeat unless you are around levels 12 or 13.

Inside of the Ice cave (the one with the floater) is an Eye protecting a chest (with the floater in it). The Eye is worth 800 experience and dies in one or two hits.

The REAL side puzzle game
Once you get the ship, the real method (at least the only one I have found that works) is to hold down the X button and press O about 55 times or so. Many people and sites will say that you tap both buttons, but this hasn't worked for me. Once you bring up the slide puzzle game you can get money, potions, antidotes, and gold-needles... all without leaving the ship or risking your party's health.
(My best record is 34.07sec and I have been able to buy whatever I want in every town I have traveled to... so far)

Speed up Airship (FF2)
To make the Airship in FF2 fly faster, simply hit the O button about 15 times. It will fly twice as fast. To alternate between fast and slow afterward, just push O one time to switch.

Hidden card game in FFII
If you hold the X button and press the O button 15 to 20 times while you are on the Snowcraft, you can play a card matching game.

Action-Cancel in FF2
A commonly used trick in FF2 is to perform an action, then cancel it during a battle. That makes leveling up weapon skills and magic spells very easy. For example, if you want to raise your weapon level, select Fight, select a target, then cancel the action, and do it over and over again. Magic is done in a similar manner. Select Magic, then the spell, then the target, then cancel it. Depending on the monsters being fought, the amount of times you will have to do this varies, but it could be anywhere from 50 to 100 times per level. Note that once you obtain 100 skill points for your current spell/weapon level you can no longer gain any more in that battle. Also changing the cursor settings to Memory might make leveling magic up this way easier.