Here is the full list of all 31 Sid Meier's Starships achievements.
Control 5 Galactic Wonders
Win a Wonder Victory
Defeat the Pirate planet
Repair a critical hit on a ship in your fleet
Win a mission by bringing your flagship to a warp portal
Win the game as Supremacy
Win a Population Victory
Take over an enemy's home planet
Research a tech to level 4
Create a warp portal between two planets
Upgrade the population on an earth-like planet
Destroy a ship with a torpedo
Use a jump gate during battle
Bring a planet fully into your empire
Score a critical hit on an enemy ship
Win a mission by bringing the colony ship to a warp portal
Upgrade the lasers on your ship
Have a ship in your fleet move 7 or more hexs in one turn during a battle
Travel to another planet
Win the game as Purity
Win the game as Harmony
Defend your home planet
Give away a tech through negotiating with an enemy
Raise a planetary improvement to level 5
Research 5 techs to level 4
Have 4 or more fighters on the map at once during a battle
Upgrade the stealth module to level 6
Increase the population on a planet to 10
Start a game using a save game from Beyond Earth
Win a Science Victory
Win a Domination Victory