Here is the full list of all 35 Oozi: Earth Adventure achievements.
Complete 10 arcade levels
Kill 25 enemies using bombs
Complete 5 arcade levels
Kill 10 enemies using a falling rock
Beat 10 challenge levels
Kill 5 enemies by bouncing from one to another
Find all 100 secret stars
Complete a story level without taking any damage
Complete a story level in less than 5 minutes
Beat the final boss
Complete the game on normal difficulty or higher
Complete the game on hardcore difficulty
Complete the game on easy difficulty or higher
Score total of 200,000 points in story mode
Punch 100 enemies
Kill 100 enemies using stomp
Beat the third boss
Collect 15,000 stars
Kill a hedgehog
Find 50 secret stars
Beat the second boss
Kill 10 enemies using another enemy
Score total of 100,000 points in story mode
Complete a story level without killing any creature
Collect 8,000 stars
Beat the first boss
Collect 1,000 stars
Find 10 secret stars
Complete a story level with perfect score
Beat 22 challenge levels
Beat all 44 challenge levels
Complete all challenges with 3-star rank
Complete an arcade level with 0-star rank
Complete all 20 arcade levels
Complete all arcade levels with 3-star rank