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