Here is the full list of all 78 Shortest Trip to Earth achievements.
Reach sector 3 to unlock
Have 7 serious (red) malfunctions in the ship in the same time
Reach sector 2 to unlock
Reach sector 4 to unlock
Reach sector 5 to unlock
Reach sector 7 to unlock
Reach sector 9 to unlock
Win the game in hardcore mode while starting from sector 1
Win the game with Tigerfish while starting from sector 1
Win the game with Nuke Runner while starting from sector 2
Win the game with Fierce Sincerity ship, starting from sector 1
Win the game with Rogue Rat ship, starting from sector 1
Win the game with Pumpkin Hammer ship, starting from sector 1
Win the game with Atlas ship, starting from sector 1
Win the game with Warpshell ship, starting from sector 1
Win the game with Endurance ship, starting from sector 1
Have 18+ crew
Have 18+ drones
Complete a sector while having only 1 crewmember
Have 1 crew and 1 pet
Have 80+ hitpoints
Have 70+ shieldpoints
Have 10+ core module slots
Have 4+ integrity modules installed
Have 5 weapon slots
Have 6 weapon slots
Have 7 weapon slots
Have 8 weapon slots
Collect 50-70 fate during a run
Collect 71-110 fate during a run
Have 123+ fate
Destroy at least 30 hostile ships in a single run
Survive 20 battles in a single run
Survive 30 battles in a single run
Die 77 times
Die 50 times
Die 40 times
Die 33 times
Die 25 times
Die 17 times
Die 14 times
Die 9 times
Die 7 times
Die 5 times
Die 3 times
Die 2 times
Die while having 2+ nukes
Die while not having a warpdrive
Die while not having engines
Die while having 0 fuel
Have 10 garden modules installed
Have 12+ crew and/or pets in cryosleep
Have 1000+ fuel
Have 25+ exotics
Own 8+ weapons
Own 6+ capital missiles
Have 60+ power
Have +30 total organics production
Win a ship-to-ship combat with 1-3 hitpoints
Escape from combat with 1-3 hitpoints
Own 5+ artifacts
Win the game with less than 10 max HP
Win the game with less than 5 max HP
Own 10+ weapons
Own 8+ capital missiles
Have 80+ power
Have +80 total organics production
Have 1 core module slots installed
Have 2 core module slots installed
Have 3 core module slots installed
Have 5 core module slots installed
Have 70+ hitpoints
Have 60+ hitpoints
Have 50+ hitpoints
Die while having an unused nuke
Win the game with Bluestar ship
Win the game with Easy Tiger ship