To be able to unlock other settlements, you will need more experience which you can receive by playing/escaping. For these achievements, starting the level should be enough:

Planet AE-23-M
Successfully arrived Planet AE-23-M. Let's start a new thriving Settlement!



Planet GL-94-G
Successfully arrived Planet GL-94-G. Let's start a new thriving Settlement!



Moon ER-82-G (3)
Successfully arrived Moon ER-82-G (3). Let's start a new thriving Settlement!



Planet TE-82-C
Successfully arrived Planet TE-82-C. Let's start a new thriving Settlement!



Planet QV-45-M
Successfully arrived Planet QV-45-M. Let's start a new thriving Settlement!



Planet HL-54-E
Successfully arrived Planet HL-54-E. Let's start a new thriving Settlement!



Planet TX-11-B
Successfully arrived Planet TX-11-B. Let's start a new thriving Settlement!



Asteroid A-9044-C
Successfully arrived Asteroid A-9044-C. Let's start a new thriving Settlement!



Lifeform R-MRJ-364
Successfully landed on Lifeform R-MRJ-364. Let's start a new thriving Settlement!



Spacebase 11-O-47
Successfully landed on Spacebase 11-O-47. Let's start a new thriving Settlement!

You can use Spaceship Building Mode to get these achievements quickly.
See You in the Stars
Escape a planet in first try

• You will get this one without even trying. If you cannot get it, you can check "Rocket Scientist" section.

Gotta Go Fast
Escape a planet in 15 seconds or less

• Adding more thrusters will provide more speed and will help you get this achievement.

Lost in Space
Have 3 consecutive unsuccessful spaceship launch attempts in a single run

• This one is the hardest achievement in this section. First of all, do not get the settlers that lowers malfunction or explosion chance. Even though spaceships will be more expensive, making the ships heavy will grant this achievement. I did try this with 80 tonnes and was able to escape the planet. So, I would recommend using 100 tonnes. Do it 3 times and you will get the achievement.

Safety First
Launch a spaceship with 0% explosion chance

• Simple enough. You can recruit settlers that lower explosion chance and simply build enough parts for your ship.

The Engineer
Launch a spaceship with 0% malfunction chance

• Same logic with the achievement above.

Rocket Scientist
Launch a spaceship with both 0% explosion and 0% malfunction chance.

• You can recruit settlers that lower explosion and malfunction chance and simply build enough parts for your ship.

Not Today!
Escape a planet with 10% or less Base Integrity

• This one is about Base Defence Mode. After getting ready to escape, you can demolish the repair station and some of your turrets to lower your base integrity. Be careful not to demolish too much.

Archaeologist
Find all the artifacts before putting any pods on the planet

• Self-explanatory, get the rewards from the artifacts before doing anything. There are 3 at each planet.

Busy Bee
100 pods placed in a single planet

• Check the "The New World" achievement.

Are you Colonizin' Son
200 pods placed in a single planet

• Check the "The New World" achievement.

The New World
500 pods placed in a single planet

• 500 pods are a lot and this will take some time. Trying "Serenity", "More the Merrier", "Galactic Monopolist" and this achievement at the same run is recommended. You will need a big and calm map; first map can be useful for this achievement. I am not sure if you can use the Sandbox Mode (level 7 required) to receive this achievement.

Serenity
Get 100% negative event evade chance

• Research and build 5 Defense System Pod (requires to be at least 8th floor). Trying "The New World", "More the Merrier", "Galactic Monopolist" and this achievement at the same run is recommended.

More the Merrier
Get 15 settlers in a planet

• Increasing Inhabitancy will send you more people in a quicker manner. 15 is the maximum population. Trying "The New World", "Serenity", "Galactic Monopolist" and this achievement at the same run is recommended.

Against All Odds
Escape a planet with 4 settlers

• You can have more than 4 settlers but do not forget to send them with Space Shuttle before launching the Spaceship.

Space Plague
Have 3 settlers sick at the same time.

• Do not treat the sick settlers and wait for the negative events. Early maps are recommended, since later levels do not have this event.

It's Friday
Get all settler morale over 90%

• Accomplishing settlers' mission will increase their morale. So, you can get this one quickly. Also, you can research and build Entertainment Centers to increase morale.

Galactic Monopolist
Research all structures in a single run

• There are 27 researches in total. You can try the sandbox mode for this but I am not sure if it is doable with it. Build Observatories and Research Stations to receive research points. Trying "The New World", "Serenity", "More the Merrier" and this achievement at the same run is recommended.

Emperor Nero
Rebuild 15 burned down pods during a single run

• Lifeform R-MRJ-364 only has this event. So, I would recommend using this map.

Hold the Line
Survive 10 waves in Base Defense Mode or Asteroid Planet.

• Building a small base with lots of turrets will be necessary. Also, Repair Station can be useful. Here you can see my defense:

Efficient Engineer
Escape a planet in Base Defense mode without getting any structures exploded.

• Enemies at Base Defense mode do not explode the buildings separately. So, you will get this after escaping at Base Defense mode.

Thank you for checking my guide. If you have any suggestions or if you see any mistakes (information, typo, etc.), please let me know. Have a nice day!