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