Here is the full list of all 31 Sid Meier's Starships achievements.
Travel to another planet
Upgrade the lasers on your ship
Win the game as Supremacy
Win the game as Purity
Win the game as Harmony
Take over an enemy's home planet
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
Destroy a ship with a torpedo
Raise a planetary improvement to level 5
Use a jump gate during battle
Research 5 techs to level 4
Control 5 Galactic Wonders
Defeat the Pirate planet
Have a ship in your fleet move 7 or more hexs in one turn during a battle
Have 4 or more fighters on the map at once during a battle
Win a mission by bringing your flagship to a warp portal
Win a mission by bringing the colony ship to a warp portal
Upgrade the stealth module to level 6
Score a critical hit on an enemy ship
Repair a critical hit on a ship in your fleet
Increase the population on a planet to 10
Bring a planet fully into your empire
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