Here is the full list of all 39 Rad Rodgers: World One achievements.
Complete a level without dying
Skip a cutscene
Complete W1L1
Find a Secret Area
Defeat Mother Nature in 4 minutes
Collect a Lion Trophy
Collect all Lion Trophies
Complete W1L1 100%
Complete W1L2 100%
Complete W1L3 100%
Complete W1L4 100%
Complete W1L5 100%
Complete W1L6 100%
Complete Jungle World 100% on Normal mode
Find the Dopefish
Find all Secret Areas
Watch all the credits
Dusty Slam every kind of enemy
Complete a level without killing a single enemy.
Complete and kill all enemies in a level only using Dusty Slam
Drown 5 times
Kill every single enemy in the entire game, no enemies left alive
Push 5 enemies to their deaths
Collect all Gems in the game
Visit all huts
Complete W1L7 100%
Complete Jungle World 100% on Hard Mode
Complete Jungle World 100% on Easy Mode
Complete W1L8 100%
Defeat the first Boomer in less than 1 minute
Defeat the second Boomer in less than 1 minute
Find the Excalibat
Use the Photo Mode
Find all Hats
Unlock all Artwork
Complete Pogo Vertigo 1 100%
Complete Pogo Vertigo 2 100%
Complete Pogo Vertigo 3 100%