Here is the full list of all 24 Sun Blast: Star Fighter achievements.
Finish the first level
Defeat the Spearkon Destroyer
Defeat the Ankoladon Devourer
Defeat the heavy Devourer
Defeat the heavy Destroyer
Defeat the Ankoladon Queen
Earn Bronze Medal in Spearkon Invasion challenge
Earn Silver Medal in Spearkon Invasion challenge
Earn Gold Medal in Spearkon Invasion challenge
Earn Bronze Medal in The Abyss challenge
Earn Silver Medal in The Abyss challenge
Earn Gold Medal in The Abyss challenge
Earn Bronze Medal in Ragin Demons challenge
Earn Silver Medal in Raging Demons challenge
Earn Gold Medal in Raging Demons challenge
Earn Bronz Medal in Asteroid Frenzy challenge
Earn Silver Medal in Asteroid Frenzy challenge
Earn Gold Medal in Asteroid Frenzy challenge
Earn Bronze Medal in Morning Stars challenge
Earn Silver Medal in Morning Stars challenge
Earn Gold Medal in Morning Stars challenge
Earn Bronze Medal in Ankoladon Swarm challenge
Earn Silver Medal in Ankoladon Swarm challenge
Earn Gold Medal in Ankoladon Swarm challenge