Here is the full list of all 32 Bombing Quest achievements.
Killed 100 enemies
Kill 10 enemies charging at you
Killed 20 enemies
Kill an enemy charging at you
One of your bombs Exploded Another
Killed the first enemy
Killed 300 enemies
Three of your bombs exploded in a chain
Four of your bombs exploded in a chain
Kill 30 enemies charging at you
Kill 3 enemies with one explosion
Kill 4 enemies with one explosion
Kill 5 enemies with one explosion
Kill 6 enemies with one explosion
Kill 7 enemies with one explosion
Kill an enemy with punched bomb
Kill 20 enemies with punched bombs
Kill a boss with punched bomb
Dash 200 times
Defeat The Beastmaster
Defeat The Soul Apparition
Defeat The Bloodseeker
Defeat The Fallen Guardian
Defeat The Broodmother
Defeat The Forgotten One
Defeat The Soul Guardians
Defeat The Artificial Mind
Complete the Tutorial
Complete the Early Access Storyline
Killed 100 enemies without dying
Killed 300 enemies without dying
Kill 3 bosses without dying