Things to mention before we start:

• Achievements are awarded on any difficulty setting.
  If things are really looking grim, you can lower the difficulty.
  
• All achievements can be completed in single-player, if needed.
  Having a second player does make a few of these easier to obtain though.
  
• Hereby I'll try to keep the descriptions short and to the point.

Finish the tutorial level.

This is unlikely to be any trouble.

Beat any level in two player mode (not with an AI partner). Tutorial level does not count.

If you do not have anyone to play with, the wormhole level in the first campaign is a good way to get this achievement:

• Set up the game to use keyboard for both characters, with "action" buttons being comfortably close to direction buttons.
  
• Position the engine between two weapon stations.
  
• Position the shield to protect the opposite side of the ship
  Having a Power 1\2 or Metal 1\2 helps to cover a wider area.
  
• Put both characters on the said weapon stations.
  
• As the level progresses, keep the characters on the weapon stations, constantly firing to clear out the enemies near the engine first, and then clear the rest (when moving a character to other station no longer poses any risk to the engine).
  If you cannot multitask sufficiently, align one of the weapons to fire in a line above the engine to protect it from stray bullets, and keep it firing as you pay attention to the other weapon.

Beat any level in single-player mode (with an AI partner). Tutorial level does not count.

While the AI cannot operate the engine, it is pretty good at operating the weapon stations and the shield. Also it won't complain about having to run between the stations a lot.

Defeat Ursa Major. This boss is not too much trouble.

You can keep the ship stationary during the fight.

The bear's melee attacks are the only subject of worry - have one of the players control the shield to block them. There is a visual cue (a "shockwave" originating at the center of a paw) before an attack happens, giving enough time to prepare.

Defeat Cetus.

The boss itself presents a minimal risk, roaming around at a constant pace for the most of the fight and attempting to do "bites" at the player ship, harmlessly bouncing off if blocked by a shield. The mines that it lays, however, deal moderate damage, and you may be pushed onto them by the boss' attacks.

Having a Metal (or Power+Metal) engine helps, as it's mine-like projectiles will neutralize the mines laid by the boss, permitting to move around more safely.

Defeat Orion. A conclusion in study of hitting things with solar flares.

The boss' melee attacks are relatively slow and easily blocked by the shield.

However, periodically the boss will launch a ball of fire in your direction, which you are expected to dodge and use (hit with projectiles to create solar flares) to temporarily remove the level of ice on the boss to be able to do damage.

Halfway through fight the boss separates into two parts, making it a reasonable idea to stay closer to the middle of the level, so that you don't have to deal with the attacks of the melee part (which can only circle the outer perimeter of the level).

As with anything else related to solar flares, Power weapons are the easiest to use (fastest firerate creating most flares), and the AI is good at picking correct angles when shooting the fireball.

Defeat King Cepheus. Nobody told you that a final boss would be too easy.

Mobility is somewhat important when fighting Cepheus.

The boss periodically releases two sets of homing projectiles, which arc towards the ship at different angles, and usually cannot be blocked at the same time. Moving around, however, allows you to let the sets to combine, making them easy to block.

Similarly, when the boss summons a number of spears (?), it is preferred that you move away from your original position, since that's where all the spears will be moving towards.

While Beam / Beam 2 weapons usually pierce through enemies, they do not pierce through Cepheus' shields, making them less of a good fit in this fight.

Ultimately it takes a bit of practice to get through this fight.

Clear a level while removing all "fog of war" (as seen on the minimap).

Easier to do with the open levels like levels 1, 2 in the first campaign.

Fishing a level without taking any damage. Apparently, any damage counts.

The "Jelly Roll" ship can save you from the potential damage from collisions with terrain, but otherwise you'll have to play well.

If you are not sure about how would you get this achievement, but intend to 100% the game, you can postpone it, since getting one other achievement will also grant you this one.

Finish a level without crossing paths with your partner. That is, without touching each other.

Works with an AI partner. Isn't a tricky achievement if you pay attention.

Have two gems in all weapon stations (including Yamato), engine, and shields.

There's more than enough gems in each campaign to do this, so you only need to explore and collect them as you go.

Finish a campaign without putting any gems into any stations.

The first campaign is, logically, the best choice for this.

If playing with an AI partner, I would consider Jelly Roll ship to be the most fit for the task, since you can align the shield on the front or on one of the sides, and save yourself from constantly juggling AI between shield and weapon stations (by rotating the ship as needed).

If you unpack the gems from the presents given between the levels (and put them down somewhere on the ship), you will also get the "Gemologist" achievement in process.

Earn one of each type of ship badge.
That is, complete at least one campaign with Gumball Zero, Banana Split, and Jelly Roll.

Getting this achievement in single-player can be a tiny bit of a trouble (since the shield on Banana Split in P2's area), but the first campaign is easily doable nevertheless.

Completing the same campaign with all there ship types also unlocks a particular character.

Rescue every creature on every level. This doesn't have to be done sequentially, and doesn't have any limits on attempts - you can revisit individual levels from the campaign menu to get the missing ranks.

Finish the Ursa Major boss fight without taking damage.

Takes a bit of practice, but manageable.

If you are having trouble, try Jelly Roll ship with Power or Metal shield aligned at the front - that way one player can rotate the ship to deflect incoming attacks with the shield, while the other player (or AI) sits on the front weapon station and only needs to aim for the boss' head.

Defeat Cetus while its head is underwater.
This achievement is not particularly troublesome - to defeat Cetus while it's head is underwater, you will need to briefly push your ship into the blob of water at the center of arena to make it possible to hit Cetus while it passes through water.
Same tactics as with the regular Cetus fight apply.

Deliver the final blow to Orion using a solar flare.

Not particularly troublesome - you get the boss' health down to a minimum value, and then you only shoot the fireballs that it launches to hit it with the resulting flare projectiles.

If playing with an AI, you will need to do the last part yourself, since the AI would prefer to just land the final hit with a weapon.

Remove all of King Cepheus's shields.

The key to getting this achievement is using two weapon stations at once. Since you do not need to defeat the boss to get the achievement, you can allow it to hit the ship as much as it desires while you actively clear out the shields.

Use every station in a single level.

For weapon stations, "use" implies shooting them (or activating, for Yamato station). The radar/map station counts too.

If playing locally with an AI, you must use the stations, so this will not be obtainable during playing with the Banana Split ship.

If somehow a trouble, this can also be done in the sequence between the levels.

Pirouette 15 times in a single level.

A "pirouette" (small animation) is performed when the characters run by each other in different directions.

This can be done even in the pre-level area where you don't have to worry about anything hitting the ship as you run around.

This can also be done while playing with an AI by ordering it to move from one side of ship to the other (so that you can run by each other).

Throw a gem into a station that is being used by another player.

Nothing particular. Also works when playing with an AI.

Have 10 unused gems lying around on your ship at once. It seems like the gems need to be unwrapped from the presents for this to unlock.
Most easily earned together with "Love Is All You Need" achievement, since you get increasingly more gems for each level that you haven't used any on.

Escape a white dwarf without shooting or ramming it.

Most easily done with a Power II engine - in this case you only need to get into the star' range, fly by it without using the charges, and use all four charges in quick succession to get a boost and propel yourself out of it's reach as you start to move away from it.

Survive after being fully enveloped by a bomb trap.

Have the ship completely inside the bomb trap effect (green area) without loosing the ship. Mostly only requires rough understanding of how large the effect grows so that you can stop distancing away from it at the right time.

Kill 20 enemies with security guns in a single level.

That is, have the enemies die to security gun projectiles.

Most easily obtained on the first level of campaign 4 (Cepheus), where the triangular structure contains a plenty of security guns, the outer areas contain a plenty of enemies (stupid enough to follow you into a trap), and you may encounter the larger enemy type that spits clots of Larvo as an attack.

Have 20 cells at one time on the Metal-Beam shield.

Every projectile that hits the cells on Metal-Beam shield becomes another cell on it.

This is incredibly tricky to obtain by aligning the ship before enemies (since the enemies will inevitably run into the crystals, destroying them and themselves), but easy to obtain by aligning the ship shield to be in front of two security guns on a slight distance. As previously mentioned, level 1 of campaign 4 (Cepheus) contains a plenty of security guns.

Make two Power-Metal missiles collide.

Requires firing the missiles from two Power+Metal weapon stations and directing them to hit each other.

Can be obtained during a level or during the pre-level sequence.

Most easily obtained while controlling both players, but can also be obtained by hitting the AI's missile as it continuously attempts to activate the switch on it's side.

Another thing worth mentioning here - rescuing all creatures on "Stellar Collapse" (campaign 4, level 2) level may prove problematic.

I would highlight 3 main factors that play towards success with this:

1. An appropriate level layout. Since there are no penalties on restarting the level, you can restart as much as you want to practice and/or trying for better luck.
Optimally you would want a layout where creature locations form a path around the map, starting near your ship, and ending just before the exit point.
The fewer hazards on the way, the better.

2. A Power II engine. You do want to get around fast, and you do need that 4-step boost for that.

3. Planning. You can open up the map as soon as the level starts (the there's no "fog of war" on this level), and the timer does not deplete while the map is open. Check the layout, and have a rough idea of how you want to go through the level without getting lost.

Having a Power II or Metal II shield helps too, since you can expect to bump into things while going fast.

Also note that you can continue moving on as soon as you break a prison - no need to wait until the creature arrives to the ship (that will happen regardless). Nor do you need to clear out the enemies spawning.

Overall, takes a bit of luck and practice, but doable.

So that's it of this guide.

If I've overlooked something or something can be done better, do tell.

And have fun!