Here is the full list of all 31 FINAL FANTASY III achievements. It takes around 15-20 hours to unlock all of the achievements on Windows.
Defeated Xande and entered the world of darkness.
Obtained all weapons sealed in Eureka.
Obtained 500,000 gil.
Received power from the Earth Crystal.
Awoke Unei with Noah's Lute.
Completed 50% of the Bestiary entries.
Defeated Garuda and saved Saronia.
Collected treasure from 50% of the chests.
Obtained 100,000 gil.
Received power from the Water Crystal.
Defeated 500 monsters.
Leveled up one job to level 50.
Received power from the Fire Crystal.
Defeated 300 monsters.
Prevented the destruction of the Tower of Owen.
Completed 10% of the Bestiary entries.
Obtained 10,000 gil.
Smashed the boulder and cleared the path.
Collected treasure from 10% of the chests.
Leveled up one job to level 10.
Defeated 100 monsters.
Received power from the Wind Crystal.
Earned all achievements.
Defeated the Cloud of Darkness and restored peace to the world.
Learned all summon magic.
Leveled up one job to level 99.
Reached level 50 for all party members.
Completed 100% of the Bestiary entries.
Collected treasure from 100% of the chests.
Obtained all hidden items.
Traveled to all the map locations.