Taimumari: Complete Edition

Following the introduction/tutorial level where you are taught the simple mechanics of the game, you will end up in the town. You will experience dialogue that the game automatically puts you in. Skip through and when finished head to the left to find the white bearded man with blue clothing.
Here you will have the option of choosing levels and saving the game

Simply select "Rocketto" from the menu and you will be teleported to the beginning of the level.

Segment 1 is easy as its all platforming with minimal enemies.
Segment 2 you will be flying and will have to dodge while shooting the enemies spawning in. Holding X and dodging is your best bet. Still, relatively easy.
Segment 3 is another platforming section, except you have wings and you can fly. Near the end there is a mini boss which you have to kill to progress. Just hold your star spell attack and kill him in between his abilities.
Segment 4 you will be flying....again... and will face enemies with another mini-boss. Hold X and chrge your star spell with RB and you will make it through this no problem.
Segment 5 is another platforming area. Fly downwards avoiding enemies and spike until you reach the bottom where you head right and onto the next area.
Segment 6 is similar to flappy bird with spikes, if you have ever played flappy bird... Basically avoid spikes while flying through, and when you reach the end, prepare for the final boss fight.

I found that holding X while charging with RB to use your star attack while getting close was the easiest method. With this, you will probably lose some health, but assuming you havent died much earlier, this really shouldn't be an issue. Staying close to the boss and dealing as much DPS as possible when not attacking will progress the battle much faster.