The tutorial should explain the basic mechanics pretty well, but here's the summary: you move with WASD and there are two basic actions: shoot and drill.
On mouse and keybord your ship auto fires by default and you hold shoot to stop.

Here are a couple of things that are not apparent:

• To drill planets you only press drill ONCE, if you hold the key you will die.
  
• You can also drill meteorites and other stuff, but not enemies ;)
  
• Holding the drill button makes your ship do a Vortex, it's a special attack to kill stuff faster. You can only do this three times per run (indicated by the three 69 symbols at the top)
  
• Don't use the Vortex, you will die... almost every time I use it I die, you are not invulnerable while using it and even though you kill stuff faster you also die faster
  
• Your ship has shields that can be upgraded in classic mode, they recharge after each wave of enemies. There is no shield indicator so try to not get hit ;)
  
• If you don't shoot for a while your ship accumulates power. When you eventually start shooting again the bullets will be huge

There are no levels here, but you do unlock different game modes. Each game mode has it's own rules, but the main one is Classic mode.

Classic mode is randomized every time with different waves of baddies and bosses coming at you, and the objective like in pretty much every other mode is to get high scores.
The main difference with this game mode is that when you kill baddies you pick up these white things called star dust. Star dust can be spent on the Upgrades Shop to improve your ship, and those upgrades are permanent.

When you start a game you will see a white line at the top left corner and a little white dot that advances towards another white line in the top right corner. This indicates how far you are from your top score.

This boss has an outer layer of rock and a core in the middle that has a shield that opens and closes.
You have to activate drill to get inside, and shoot the core when it opens it's shield. When the core turns red it means it's about to die.

Things to keep in mind here:

• Avoid the rock walls when inside the core, they are instadeath.
  
• Sometimes there are 2 big things facing eachother, attached to the inner wall of the rock circle. These things shoot a laser straight to the core, avoid those at all costs. You hear a sound going down when it's about to shoot
  
• The 2 small guns that shoot little bullets don't make much damage, avoid them if you can, but don't run into anything else while doing it.
  
• On thougher versions of this boss sometimes you have an outer purple shield protecting the rock part, make sure you go through the gap when drilling.
  
• When the core dies a multiplier appears, try to stay in the middle as long as you can before drilling out to increase your multiplier.

It's a wall of rock that spans the screen vertically and has multiple cannons pointing at you.
The cannons shoot straight so concentrate fire on one of the guns first, once you kill it start shooting the other ones. The guns shoot without much warning that's why you need to make a safe corridor first.
If the wall disappears off screen hug the left border of the screen because it will ram you.
After all the cannons die activate drill and go through the wall; don't drill through the wall before then or you will die, you can drill inside a little bit but not all the way through.

Things to keep in mind:

• Sometimes the wall spits purple baddies at you, these are horrible because they block your shots. Their forcefields don't hurt your ship, so go through them and fire on the other side. You can kill these by going inside the forcefield and firing at the little thing in the middle. Be careful when there are multiple of these, since the forcefields might overlap and they might be almost impossible to kill.
  
• Sometimes there is an antenna that also acts like a shield, shoot the other guns first if this happens

This boss looks like a spider :)
Simply avoid it's bullets and shoot it's head (where the bullets are coming from). The bullets don't hurt much unless you eat a lot of them.
After shooting for a while it stops and shoots a giant laser that is a one hit KO, try not to be in front of it when it does :D The laser doesn't give much warning, but the first indication is when the bullets stop and the second is a sound that goes down before it fires. If you here the sound run...

It's a moon that has a purple shield around it that blocks all your shots (sometimes 2 shields). You can only hit it by shooting through the gap in the shields.
After you shoot at it a little it teleports (sometimes on top of you). After teleporting it shoots a laser that joins the previous location with the current one (avoid this).
After a few teleportations it moves to the middle and it starts the bullet hell phase, when it's red it means it's about to die.

The strategy for this is pretty much shoot at it and pray it doesn't teleport on top of you (and avoid the laser)

If I'm not mistaken this is called Blackhole, time slows down in the beginning. Some dinosaur heads shoot a string of bullets or some stuff that looks like fire. The fire ones are lethal (avoid). You can kill the heads, but it takes a few shots.

You can only go through here shooting the white and red bubbles, otherwise it kills you.
When you hit a white bubble it dissapears and reappears overtime. The Red bubbles explode and clear a lot of white bubbles.
Avoid the black bubbles because they are really hard to kill, make sure you shoot a path avoiding those.

Sometimes a black circle appears that changes size and rotates really fast all over the screen. It's harmless to you and will help you for a little while by killing enemies. It's random and only temporary.

When you start upgrading your ship you might notice that sometimes your ship fires a huge laser that obliterates the entire screen. This happens automatically when you collect a certain number of star dust. You can't control it so just... accept it...

The screen turns white (negative colors) and the music changes. When this happens the enemies you shoot explode into multiple spike balls, try not to shoot groups of enemies when this is happening or the screen will fill with spikes.

These are called gas giants and they are explained in the tutorial. Go through the bars to collect free star dust

This list is pretty short, a good source of information is the "Library" entry in the game's main menu. This fills over time and explains most things on the game

Thanks for reading, I made this guide because I thought it would be useful. Please let me know if something is incorrect or if you think I should add something to it.