Here is the full list of all 30 FINAL FANTASY IV achievements. It takes around 20-25 hours to unlock all of the achievements on Windows.
Accidentally destroyed the village of Mist.
Earned all achievements.
Climbed Mt. Ordeals and became a paladin.
Defeated Cagnazzo, the Lord of Water.
Defeated Barbariccia, the Lord of Wind.
Arrived in the underworld.
Defeated Rubicante, the Lord of Fire.
Obtained the Lunar Whale and landed on the moon.
Defeated the four elemental lords again.
Defeated Zeromus and restored peace to the world.
Obtained Excalibur.
Exchanged the pink tail for Adamant Armor.
Obtained all sealed equipment.
Learned all non-secret summon magic.
Found and learned all secret summon magic.
Reached level 70 for all party members.
Defeated 100 monsters.
Defeated 500 monsters.
Defeated 1000 monsters.
Completed 10% of the Bestiary entries.
Completed 50% of the Bestiary entries.
Completed 100% of the Bestiary entries.
Obtained 10,000 gil.
Obtained 100,000 gil.
Obtained 500,000 gil.
Collected treasure from 10% of the chests.
Collected treasure from 50% of the chests.
Collected treasure from 100% of the chests.
Obtained all hidden items.
Traveled to all the map locations.