You will get these achievements for as long as you just follow the storyline and complete the game.

Unknown Space
Enter the Wormhole.


Rescue
Evacuate the Outpost.


The Machine Stops
Shutdown the Raider Factory.


My god, it's full of stars
Activate the Polyhedral Artifact


No Place Like Home
This is definitely no place like home.


What lies behind doorways that he himself has sealed
Enter the unstable universe.


The End
Finish the game.

You can get most of these achievements within a few minutes of starting a new game.

Death by Misadventure
Get killed by the cargo spill.


Your first mission in the game is to clear a cargo spill (at -16,58 X -7,67). Some of it is explosive - just let yourself be destroyed by it.

Sailing to Byzantium
Meet with the Byzantium.


At the start of the game, fly from the cargo spill directly to south-west, towards the Alcyone system (-17,4 X -8,4). Fly around that system until you meet a familiar face (Byzantium is a ship from the first game).

Unfinished Business
Destroy the cargo debris after returning to Starcom space.


You won't get this achievement at the start of the game, but it's important that you don't destroy all the spilled cargo at the start (which contrasts the achievement from the first game, where you had to destroy all the asteroids). For as long as there is some cargo remaining, you will be able to get this achievement later, when you return to this universe - just destroy everything when you get back.

If you are reading this after you have already destroyed all the cargo and don't want to replay the whole game, you can fix this by editing your save file - find the persistent object "ALPHA.DEBRIS_FIELD" and change its "currentObstacles" to something greater than zero (source).

In order to get these achievements, you just need to visit as many star systems as possible and survey every planet there.

Explorer I
Complete 10 planet anomaly surveys.


Explorer II
Complete 50 planet anomaly surveys.


Explorer III
Complete 100 planet anomaly surveys.

These are different achievements that you can get throughout the game. They are sorted by the time you can get them, from earliest to latest.

Soldiers of Fortune
Bribe the Ermyr.


You will find Ermyr in one of the first systems after you enter the wormhole (I found them at 14,4 X 10,3), and you'll soon discover that they are mercenaries that will do anything for copper. But, in order to discover that, you will first need to survey a "Starliner Wreck" anomaly (found mine at 14,60 X 10,32). After surveying that anomaly, talk to the nearest Ermyr and pay them 10 Copper deal with Red Raiders in order to get this achievement.

Shield Shove
Open the shield station another way.


You will discover a space station that is protected by a shield (3,7 X 13,6) and can't be explored until you disable that shield. The main way to disable it is to land on the nearby planet that's generating the shield. But, in order to get this achievement, you need to use another way - just push the station far enough from the planet with your ship (ram it).

The Emissary
Follow the Emissary.


You will eventually discover a blue probe that's flying around different star systems (I found my first one after "The Emissary" anomaly at 10,84 X 8,64). Follow it, and you'll discover that it will soon enter a small wormhole. Fly into that wormhole to get the achievement.

A Diplomatic Mission
Find the Missing Diplomat.


You will get it during a mission "The Missing Diplomat" from the Nimion race (which is located in star systems to the west, e.g. 46,5 X 22,5). The diplomat can be found at "The Escape Pod" anomaly (44,63 X 16,45).

Tomb of Ock Juk
Solve the mystery of Ock Juk's tomb.


It's an anomaly called "Colossal Caves" on one of the planets (19,51 X -2,60). There, you will need to complete a small labyrinth and obtain Ornate Scepter. After that, talk to one of the Legion Ermyr ships (they can be found at 31,2 X 10,3) and agree to sell it for 200 copper.

Bones of the Dead God
Find the Bones of the Dead God.


They can be found during the anomaly "Bone Pyramid" on one of the planets (12,32 X 2,69). Before that, the anomaly "Sandstorm" (10,43 X -6,39) will give you an altar which, 10 days after you pick it up, will give you a hint to the bones' location (if you can't find it at my coordinates).

Lost in Space
Find a Freighter off the beaten path.


It's an anomaly called "Lost Freighter" on one of the planets (mine was at -25,47 X 19,50).

Fear the Reaper
Kill the Void Reaper ship.


Void Reapers are pirate ships that travel in the deep void, far away from any star systems, and ambush unsuspecting prey. They place Void Disruptor traps that significantly slow you down, so, if you manage to find such a trap, a Void Reaper ship is likely nearby. Don't destroy the trap before you destroy the Reaper, otherwise it may run away. Also I would recommend doing this achievement late in the game with an upgraded ship, because Void Reapers are somewhat strong.

I have managed to find a Void Reaper and its trap at around 14,6 X 45,3, but there are likely more locations deep in the void where they can be found.

You can only get these achievements at the very end of the game.

Pale Blue Dot
Visit Earth.


After you return to your home universe as a part of the mission "Home Again", and use a long flinger to fly towards the Research Lab, you need to backtrack a little to one of the star systems you flew by. The system in question is Sol, and it's the nearest system to the southwest from the Research Lab (0,4 X 0,5). Once in Sol, just find and scan Earth.

Secret Ending
Follow the Generation Ark into the unknown.


In order to get this achievement, you will first need to complete "The Talonian's Task" mission. After it's complete, talk to your commander as a part of "Home Again" and tell him you're ready to head home. But, instead of flying through the wormhole to your universe, fly into the "immodest singularity" (-37,2 X -28,2, or find "singularity" in your logs). Once inside, approach the "white hole" and hang around it for a few seconds, until you get a message ending with "I guess we're not going home?". Then fly into the white hole to unlock this achievement and get a unique ending sequence.