These achievements cannot be missed as you play through the levels, there are so many, that this guide is pretty pointless. The levels themselves are not too hard to complete with some patience.

Complete the story.
start the journey
clear level m1


meet the Earth
unlock the Earth


meet the Venus
unlock the Venus


meet the Mercury
unlock the Mercury


near the Sun
unlock the Sun


meet the Sun
clear all normal levels


Moon rising mode is unlocked upon completion of the main story.
dark moon rises
clear level sr10


meet again the Mercury
unlock Mercury_r


meet again the Venus
unlock Venus_r


meet again the Earth
unlock Earth_r


brighter than sunshine
clear all reversed levels


These achievements are more than likely to unlock along the way of the story.
space crashers
destroy more than 500 blocks


super moon boy
jump over 200 times

gravity free
clear a level without jumping


This one you will probably get on the way. If not, I got it on the level 'e4'. But it's confirmed possible on m2, e1, e2, e3, e4, e6 and e7 at least.

still alive
warp 10 times from top to bottom at once


For this one you can just move away from a platform and wait till you loop the screen 10 times. Confirmed by K3W3L. Thank you!

faster than light
clear level without using drop of light


The trick with this one is that you have to complete the level while holding a drop of light.
I got this one on the level ''s1", but you can probably get it sooner:

Jedo made a video of getting this achievement on the level 'v1':