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