Here is the full list of all 30 Ancient Islands achievements.
Finish the third level
Finish the sixth level
Finish the ninth level
Collect 2500 gold coins from slain enemies
Collect 5000 gold coins from slain enemies
Collect 10000 gold coins from slain enemies
Slay 500 gnolls
Slay 1500 gnolls
Slay 3000 gnolls
Deal 10000 points of magical damage
Deal 25000 points of magical damage
Deal 50000 points of magical damage
Collect 10000 trees
Collect 25000 trees
Collect 50000 trees
Collect 10000 stone
Collect 25000 stone
Collect 50000 stone
Deal 2000 points of damage by casting spells
Deal 4000 points of damage by casting spells
Deal 6000 points of damage by casting spells
Defeat 2000 enemies
Defeat 5000 enemies
Defeat 10000 enemies
Trade 5000 resources on the market
Trade 10000 resources on the market
Trade 15000 resources on the market
Build 25 archer towers
Build 50 archer towers
Build 100 archer towers