Here is the full list of all 17 Bombing Bastards achievements.
Complete the first level
Complete all Fairy levels before the Fairy Boss
Complete a level with 3 stars
Complete 5 levels with 3 stars
Complete 10 levels with 3 stars
Complete 20 levels with 3 stars
Complete all 30 levels with 3 stars
Defeat the Fairy Boss
Defeat the Fairy Boss with 3 stars
Defeat the Ice Boss
Defeat the Ice Boss with 3 stars
Defeat the Wind Boss
Defeat the Wind Boss with 3 stars
Defeat the Fire Boss
Defeat the Fire Boss with 3 stars
Defeat the Spark Boss
Defeat the Spark Boss with 3 stars