Currently there are 14 achievements available. Of course, there will be more, perhaps sooner or later. [0.94.32]
Since the game only came out, some of them don't function properly. Keep that in mind.
TIP: If you still haven't completed the story, you should. As you will see all levels where you can easily get the achievement.
Also sorry for big icons, this is their original size lol
List:
-Wheeeee!
-Not so fast!
-Helping Hand
-True Killer
-Prospector
-Gunslinger
-End of the line
-Scan-Ball kill
-Flying rock
-True Miner
-Just hanging around
-Indestructible
-Completely Shafted
-M-M-M-Monsterkill

Use the grappling hook as a zip-line.
You will get this opportunity at second level, AKA Lost.

Kill the ceiling tentacle before it kills you on your first encounter
Pretty easy to get, slowly move to the monster and when it starts moving out, shoot as fast as you can.

Let another team member use your rope to climb
Again, you will get this opportunity at second level. You can't miss it.

Kill 100 monsters
Self explanatory, but if you need to farm, go to the challenge mode, set find data disk and set monsters to "What?" difficulty.

Find a hidden cave using Terra-Scanner
Currently, this achievement binded to level 2, so it won't work on other levels. But this will be fixed soon.
This one is a bit tricky. You need to scan fake cave first (which on the left) and then scan correct cave, breach it, and enter. Otherwise you won't get it as i did, because i scanned correct one.

Kill 20 enemies with your gun without dying
Self explanatory. Don't die.

Rappel down the full length of your grappling hook's line.
You can get this achievement in multiple levels. Especially on the last one, it will be easy. You also can get this without trying.

Kill 3 enemies using one Scan-Ball
This one is a bit challenging, you have to find a cluster of enemies (they usually spawn in big empty areas) and kill them with one scan ball. I got this achievement playing on Intense Exploration and actively using scan ball. You can try in challenge mode by setting monsters amount to "Insane" or "What?"

Fall from 50 meters or more
I got this achievement on level 9, AKA Unstability. Just move crane as high as you can, and fall from it.

Drill 100 meters of tunnel using the Tunnel Boring Machines
Could be a challenging one. You can use a bug, on a Deadly Sphere level, use your machine and drill through a tunnel, when you finished, continue moving you'll notice ladder,
continue, and start to drill.
Otherwise, you can just replay these missions until you will get these achievements.
It will turn out into this:

Kill 5 enemies in a row while hanging on the grappling hook
Could be a challenging one, easy to get on Intense Exploration+, where you have a lot of enemies.
Empty big areas contain a lot of monsters, so when you will get near this area, get on a rope and start shooting, (Don't run out of ammo too fast), and aim precisely. Also you can try same tactic on max diff challenge mode.

Complete a level with high enemy presence without dying (must be 50+ enemies)
You can try to get this achievement in challenge mode clear out the area. Set monsters to 50 (it will be 60 in the mission) and try to kill everyone without dying. You have movement detector, so it will be easier to avoid monsters. After killing all, you will get the achievement.

Ride a elevator from top to bottom non-stop.

In order to get this achievement, you need to find 4-floor elevator
I don't remember how i got this achievement, but in order to ride it all the way up, you need to hold Shift + W or S. You can try this with every elevator on the mission or easiest, on challenge mode. If still don't, try to replay all missions and ride every elevator, eventually you will get it.

Kill 1000 monsters
I easily got this achievement by playing on Intense Exploration. Just farm monsters as you want, on challenge mode, or story.

That's all, for now. There will be more in the future, but now we will have to wait. Left with questions? Ask in the comments and perhaps i will answer.