Good Egg Galaxy
First Planet: Go ahead and fly to the Good Egg Galaxy to get your adventure underway. The first planetoid here will have a hidden area: if you head to the dark underside of the planet, you can find a golden pipe that will lead into the house on the upper side. Follow the trail of musical notes that appear after hitting the question mark to reveal a 1-Up.
Second Planet: On the second planetoid, find the five shards of the star, put them together, then launch yourself off to the third planet.
Third Planet: Kill the enemies here by chucking Star Bits at them with the B button, then hitting them. When the Sproutle Vine sprouts, hit the large coin to cause a number of other Star Bits to appear, then climb the vine by spinning around its base.
Fourth Planet: Make your way past the rolling blocks until you reach the huge man-eating plant. In order to kill it, you need to move behind the green plants nearby, position them so that they're between you and the plant, then spin into them so that they hit the plant. It'll poof away, revealing another Sproutle Vine.
Fifth Planet: There's a 1-Up hidden behind the stone blocks, so grab it before heading up top and revealing the Launch Star.
Sixth Planet: Your first boss fight occurs here, against the massive Dino Piranha. In order to hurt him, you need to spin near the bulb that comes out of his large tail: doing so will cause it to smash into his head. It can be tricky to time it right, though, since he'll try to chase you down. If he charges at you, don't run away: he'll catch you. Instead, move off to his side and run right to his behind, then spin near his tail again. When he loses sight of you due to your movements, he'll slow down a bit, making this a bit easier than it seems.
Keep in mind that you can shoot star bits at the small flowers on the ground here to generate coins to heal yourself.
A Snack of Cosmic Proportions
When the Dino Piranha has been defeated, you'll return to the Terrace Observatory automatically. After the lesson from the Luma, head down to the next portion of the Good Egg Galaxy.
First Planet: This is the same first planet as the one you saw before. Be sure to collect all the star bits here that you can before heading to the platform on the dark side where a Luma will shoot you off to a new planet.
Second Planet: Again, find the enemies, kill them, grab their star bits. The Launch Star will be inside a crystal.
Note that there will be more Launch Stars in the path of your flight here; if you want to use them, simply attempt to spin in mid-flight, and you'll be shunted off to another planet. There are three main planets here, including the Second one. If you travel from the second, to the third, to the fourth, you'll find a fat little Luma that asks you to feed it 100 star bits. Stand underneath it so that the counter appears, then throw as many star bits as it takes into its mouth. That will cause a new planet to appear, as well as a new launch star.
Fifth Planet: After feeding the 100 star bits to the Luma and moving on, you'll land here. Keep in mind that you can kill spiky plants by spinning near one of the green balls; that will unlock a 1-Up for you here. Slowly but surely make your way to the top of this planet to reach another launch star.
Sixth Planet: This central planet is covered in a crystal window. Eventually you'll find a crystal that you can crack to head inside, where a number of gravity switches will keep you guessing. Typically there will only be one path to take, but watch out for platforms that raise and lower; if you stand on them when they completely raise, you'll often be pushed into another gravity field and fall off.
Eventually you'll reach the top of this planet and find another launch star. Ride it out of the window.
Seventh Planet: Find all of the star pieces here to load up a number of smaller blue stars. Press A while you're locked onto them to float up towards them. In order to reach the Power Star, you'll need to lock onto a blue star, then lock onto one on the far side of the Power Star, and continue to do this until you eventually intersect with the Power Star, which will bring you back home.
King Kaliente's Battle Fleet
First Planet: In order to move on from this planet, head to the underside, enter the red pipe, go through the house, and you'll find a launch star on the roof.
Second Planet: Use the coconuts here to destroy the spiked plants; one of them will have the launch star inside of it.
Third Planet: The giant Pokey here has the launch star inside of it. Roll a coconut over to his base, spin it into him, then spin his head when it plops into the sand to reveal it.
Fourth Planet: This planet is somewhat optional, since you land directly on the launch star leading to the fifth planet. If you get down and head down the pipe, though, you can hit the coin to become invincible Rainbow Mario. That'll let you kill the goombas and steal their star bits before moving on.
Fifth Planet: You come out onto the underside of this planet. There's a number of coins here to collect, but what you want to do is get onto the upper side, avoid the Bullet Bills that come your way, and smash the crystal that hides the launch star.
Sixth Planet: At the outset here, use your spin attack to throw the coconuts back at the launcher enemies across the way. When they're both dead, the bridge will lower. If you want to get the coins on the upper levels of the first ship here, use a launch jump: hold down the Z button to crouch, then hit A to jump extra high.
Make your way to the deck of the second ship to find the launch star.
Seventh Planet: The boss fight against King Kaliente occurs here. He's not so bad, though; all you really need to do is wait for him to fire coconuts at you, then spin them back at his face. The first one will hit him straight on, but the second and third will require you to hit them back multiple times. When you hit him in the head three times, he'll pop out the Power Star. Collect all of the star bits and coins on this planet before nabbing it and returning to the Observatory.