• 
  Space Turtle I

  Get hit 10 times without dying
• 
  Space Turtle II

  Get hit 15 times without dying
• 
  Space Turtle III

  Get hit 20 times without dying

Let enemy bullets hit you, but not too many. Powerups, especially REPAIR, replenish health, letting you "eat" more bullets without dying.
DIE HARD is the corresponding badge.

• 
  Action Hero I

  Fire 5000 bullets
• 
  Action Hero II

  Fire 20000 bullets
• 
  Action Hero III

  Fire 100000 bullets

Keep playing the game, keep shooting...
GUNNER is the corresponding badge.

• 
  OmNomNom I

  Pick up 50 powerups
• 
  OmNomNom II

  Pick up 100 powerups
• 
  OmNomNom III

  Pick up 500 powerups

Keep playing the game, keep getting powerups...
POWERHOUSE is the corresponding badge.

• 
  Death Dealer I

  Kill 500 enemies
• 
  Death Dealer II

  Kill 1000 enemies
• 
  Death Dealer III

  Kill 10000 enemies

Keep playing the game, keep killing enemies...
KILLER is the corresponding badge.

• 
  Mr. Scott I

  Absorb 50 bullets with your shields
• 
  Mr. Scott II

  Absorb 100 bullets with your shields
• 
  Mr. Scott III

  Absorb 200 bullets with your shields

When a ship respawns or activates its SPECIAL WEAPON, it gets a temporary SHIELD. While it lasts, you need to catch bullets.
BUBBLE MAN is the corresponding badge.

• 
  El Satanico I

  Kill 100 enemies in melee combat
• 
  El Satanico II

  Kill 250 enemies in melee combat
• 
  El Satanico III

  Kill 500 enemies in melee combat

You need to hit enemy ships with your ship. That deals a lot of damage to the ship, unless you're covered by a SHIELD.
LUCHA LIBRE is the corresponding badge.

• 
  Bomb the Bass

  Kill 1000 enemies with your special weapons

Every ship has its SPECIAL WEAPON. You need to kill 1000 enemies with them, not with regular bullets.
It can be easily grinded on the far right side of level "4 Unfinished Business" (you need to go past the huge ship STEEL SKULL). Wait for bombs and small enemies to appear and shoot them all with "S-104 Dart"'s SPECIAL WEAPON.
MEGABLAST is the corresponding badge.

• 
  Facepalm

  Die in less than 30 seconds
• 
  Double Facepalm

  Die in less than 15 seconds
• 
  Triple Facepalm

  Die in less than 5 seconds

Start level "1 Pilot" on difficulty "4 Pure" with "S-116 Wasp" and go right into a mine immediately...
Of course, these conditions aren't obligatory. It's just what I did.

• 
  Nostalgia

  Complete a campaign level using a retro filter

Activate any filter in the main menu, i.e., GFX must not be Modern, and complete a campaign level.

• 
  Tourist

  Complete the campaign at tourist difficulty
• 
  Young Gun

  Complete the campaign at young gun difficulty
• 
  Arcade Master

  Complete the campaign at arcade difficulty
• 
  Panzerotto Prophet

  Complete the campaign at “pure” difficulty

To get these achievements you only need to complete the last level "6 The Storm"!
Completing it on difficulty "4 Pure" is still very tough, but there is this strategy:
Use "S-104 Dart". Its speed will let you pass the groups of 3 ships that appear on your way without any trouble (with slower ships you can't pass them before they "materialize" and are likely to crash). So yes, just dash on HYPERSPEED for most of the level, slowing down only when you need to surpass some difficult obstacle. When you're almost at the end, you must slow down and clear all the enemies, especially cannons. Proceed to go through the spinning gates. The first one must be passed carefully as usual, the 2nd and 3rd ones are possible to clear of cannons completely, so I like doing that.

• 
  Roasted Abruzzese Skewer

  Complete the campaign with the Dart
• 
  Insect Man

  Complete the campaign with the Wasp
• 
  Heavy Duty

  Complete the campaign with the Mule
• 
  You don't have to be pretty to kick some ass

  Complete the campaign with the Overseer

To get these achievements you only need to complete the last level "6 The Storm"!
Completing the last level on difficulty "1 Tourist" is easy with a good ship. "S-185 Mule", however, is severely underpowered, so you will need a lot of patience in the final part...

• 
  The Most Powerful Man In The Universe

  Be one of the top players in the campaign leaderboard

Play the campaign well, preferably on "2 Young Gun" or "3 Arcade" difficulty, shoot all the things, pick up powerups, and you should get the achievement.

• 
  Immortal (kind of)

  Be one of the top players in the shield nebula leaderboard
• 
  Piracy is not a problem

  Be one of the top players in the Lighthouse leaderboard

"1 Last Line of Defense", "3 The Lighthouse"
Play these survival levels well, preferably on "2 Young Gun" or "3 Arcade" difficulty, shoot all the things, pick up powerups, and you should get the achievements.

• 
  In less than 12 Parsecs

  Be one of the top players in the Frozen Canyon leaderboard

"2 Ice Canyon"
You don't need to hurry too much, give it a few tries to see your pace. I'm not sure if you even need to kill that CYCLOPS on the right.
"4 Pure" difficulty can boost your points greatly while changing almost nothing in regard to difficulty. I recommend "S-116 Wasp" here (yes, it's not the fastest, but isn't that better?)

• 
  Flawless

  Complete a campaign level without getting hit or sufferig damage

For this, obviously, a ship with good armor is pointless. Ships with good firepower ("S-116 Wasp", "S-104 Dart") will let you get rid of enemies quicker, giving them less chances of harming you. The level "1 Pilot" seems best for doing this achievement.

• 
  Diehard I

  Survive 5 minutes in survival mode
• 
  Diehard II

  Survive 10 minutes in survival mode
• 
  Diehard III

  Survive 15 minutes in survival mode

This one can be easily sacked on the survival level "3 The Lighthouse". Clear almost all of the first wave, clear the PIRATE SHIP of all its gun parts, but leave it alive (don't damage its main part). Then you can proceed with going back and forth through the map only having to dodge the mines that appear (and, occasionally, the PIRATE SHIP) for 15 minutes...

• 
  Dodge This!

  Play 5 minutes without getting hit

You could do this by playing the level "1 Pilot" very carefully (and making sure not to finish it before 5 minutes run out), but it should also be possible to combine it with survival mode level "3 The Lighthouse".
NIMBLE is the corresponding badge.

Originally posted by supremebloodorange:

...can be easily achieved by playing the last level of the campaign "6 Storm" and staying at the beginning of the level for 5 minutes. Enemies will not attack nor move toward the player if the player stays at the start of that level.