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