Most of the Achievements in this game are pretty easy to get, but I noticed some of them might come across as difficult or vague for more casual players, so here's some help.

Cleared the stage leaving 6 magics. (Arrange Mode)
This one is a cake walk and I'm sure most players get it on accident, beat the stage with 6 magic bombs in-hand, the game hands them out like candy so it shouldn't be a problem

I shot Jizo and defeated it. (Arrange Mode, Time Attack Mode)
Jizo is the metal vase that gives you fairies, crystals and other stuff that gives you points, IIRC you get this achievement if you destroy Jizo BEFORE it has time to get a fairy for you.

Defeated the boss using magic. (Arrange Mode)
Get the boss down to low health and then use a magic bomb on it.

Clear the first page of Arrange Mode with no mistakes! (Regardless of remaining machine, character, or difficulty level)
Clear Arrange Mode 1 and 2 with no mistakes! (Regardless of remaining machine, character, or difficulty level)
Clear Arrange Mode 1 to 3 with no mistakes! (Regardless of remaining machine, character, or difficulty level)
These are pretty similar so I'm grouping them together, you need to clear levels 1 through 3 without getting hit so you can get all 3. Using barriers to protect yourself trivializes these achievements, you can also play on Easy mode to further remove any challenge!

Cleared the arrange mode with silk. (stock, character, difficulty, continue, no questions asked)
Description says it all, go through arrange mode with Silk, you're more than welcome to credit feed. You can play as Silk once you beat Arrange mode as Cotton, just go into options and switch characters.
Silk is the only other character that gets an achievement for beating arrange mode, Appli and Needle don't get their own achievements.

Defeated the boss using a fairy. (Arrange Mode)
Get a boss down to low health and hurl Silk & friends at it, using the Fire Fairies magic also works.

Special strong enemies character has appeared. (Arrange Mode)
This achievement is extremely vague on what it's talking about and is one of the main reasons I'm making this guide.
You need to wait around 3-minutes in a boss fight, once you reach that time-frame some weird bat enemies should start spawning, unless you planned on milking bosses for score it's unlikely you would get this achievement normally.

The boss was defeated by the barrier's body.(Arrange Mode)
Get a boss down to low health and then equip the barrier, after that you need to fly into the boss to deal contact damage.
It's worth mentioning that the damage dealt with the barrier is pitiful and you'll likely end up knocking Cotton out if the boss doesn't die.

I shot a black crystal and destroyed it. (Arrange Mode, Time Attack Mode)
This is one of the hardest achievements in the game just from how durable black crystals are, as soon as you see a crystal on-screen try to turn it into a black crystal immediately, shooting it up close and using magic bombs can speed this up, keep doing this until it becomes black and inevitably breaks.

You have won all the trophies!
Self-explanatory, the "trophies" part is a left-over from PSN.