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