Here is the full list of all 15 Heroes of Havoc: Idle Adventures achievements.
Unlocks the ability to upgrade your heroes' rarity
Unlock Timed 3 - The Green Meanies
Unlock Trial 3 - Only Human
Unlock Timed 2 - Dungeons and Dragons
Unlocks the mysterious power of Relics
Unlock Trial 2 - Rise of the Robots
Unlocks the ability to upgrade your equipment
Unlock the Visting Merchant
Unlock Crusade Mode
Unlock the Chest in the Main Menu
Unlock Elite Campaign
Unlock Timed 1 - The Hunted
Unlock Trial 1 - The Impvasion
Unlock Idle Mode - Heroes will continue fighting without you
Unlock the competitive PVP modes by opening the Dark Portal