Sakuya
In the 5th screen of "Power of reel", talk to Sakuya(Q) and complete the minigame, started by selecting "Let us start!". Completing the minigame will give you her picture, an achievement, and gold.
Flandre
In the 6th screen, bounce on the shielded fairies (Jump, then Down + Attack) and complete her minigame. This rewards you with her picture, an achievement, and the "Guardian gem" item.
Reimu
After completing Flandre's minigame and obtaining the "Fairy punch" ability, you must now go to the 7th screen, in which after going up the steps you will find Reimu on the rooftop. Talk to her, then punch her, and repeat this until you get her picture. This also rewards you with the secret achievement.
Kosuzu (Non boss)
After obtaining the gold in the 4th and 5th screens, (in the 4th, probe for fake floors by holding the down key and spamming jumps, and in the 5th, just do the Sakuya challenge) go to the 4 fairies on pedestals on the 6th screen and go under the first, pressing down and jump to go through the fake floor. Then interact with all of the walls (press Q), and walk right until you see Kosuzu, then talk to her. This rewards you with her non-boss picture.
Kosuzu (Boss)
(Thanks to FluffeyPanda's world on youtube and stonebeach on steam!)
When in the bossfight, you must counter her attacks according to the scrolls 3 times (The lunge should be countered by using your skill {fairy punch doesn't work}, the double slash should be countered by pressing the attack button twice {or spamming it}, and the downward slash should be countered by attacking Kosuzu while she is rising up to attack)
Doing so correctly will make a picture of her pop up on the screen and give you the "Lost intent to fight!!" achievement.

First quest
Simply boot up the game.
First blood
Simple as defeating Kosuzu.
Good Guy
On the 7th screen, traverse normally until you see a 2-way platform. Go through it, then talk to the small fairy on the right. Complete her quest and you will get the achievement as well as the "Orb of wind".
Deadly Banana
Also on the 7th screen, traverse normally until the end of the screen, where you should see a platform with 3 fairies. Instead of jumping up to go to the next screen, walk straight until you go through a wall. Talk to Tewi and give her 2 gold, which gives you the "Secret plan" item. Then go to the Kosuzu boss, and you'll get the achievement once she slips on the banana.
Reel of Wind
On the first screen, bounce on the blonde fairy (Down + attack + jump), then jump and dash again. This will get you to a secret room with a key, pick it up and go directly to the Kosuzu boss fight. After defeating her, jump to the 2-way platform near the slime, and interact with the wall (Q). Then just pick up the Reel of Wind.