Here is the full list of all 31 Sid Meier's Starships achievements.
Control 5 Galactic Wonders
Win a mission by bringing your flagship to a warp portal
Win the game as Harmony
Win a Population Victory
Take over an enemy's home planet
Create a warp portal between two planets
Research a tech to level 4
Upgrade the population on an earth-like planet
Bring a planet fully into your empire
Use a jump gate during battle
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
Repair a critical hit on a ship in your fleet
Destroy a ship with a torpedo
Score a critical hit on an enemy ship
Upgrade the lasers on your ship
Travel to another planet
Win the game as Supremacy
Win the game as Purity
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
Defeat the Pirate planet
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 Wonder Victory
Win a Domination Victory