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