As aforementioned, the star chips are scattered on the small planet. Look in crystals, on the grass, above structures and beside them as well. You'll be able to find a 1-up on top of one of the rocks. Watch out for the Goombas and Falling Meteors, as they'll deplete your life or might knock you into the Black Hole!
Once you find all of the chips, a launch star will appear. Soar onto the next planet where you'll see a pink crystal thing jutting out of the ground. If you spin attack it, it'll cause all of the Goombas near it to topple over. Go to the top where the Luma will explain that a Goomba has a key. Kill the lighter colored Goomba to get the key and free the Luma. It'll transform into a sling star (a less powerful launch star), and you'll then be given access to the next planetoid.
Head to the top of this area and watch out for the moving platforms (if you touch either the starting or ending point of one, you'll be shocked). At the top there's a colossal Goomba that can be killed after you spin near the pink crystal. It'll topple over, and you'll then be allowed to kill it, thus giving you the next key. If you're looking for Star Bits, then there are a few at the top, but if not, then go down into the pipe that's just opened. Make all of the switches flip blue by running over them. Make sure not to do so again, cause they'll switch back to yellow. Dodge the moving platforms and Goombas, and then collect your first Grand Star!
When you pick this mission, head up the building where Rosalina will be. After talking with her, a red Luma will ask you to find all 100 purple coins on the planet that you're on, and then explains the red star (If you are doing this mission again, then only the red Luma will be here). Take the star, then jump in the air and spin. You'll start to fly! Collect all of the coins in the air here, then head over to the stones jutting out of the ground and collect all of the coins on top of them. If you spin while in mid-air while near a coins, it'll gravitate toward you; so that's a plus!
Just a note - 70 of the coins will be near the surface, while thirty will be in the air and will require the red star suit. If the suit ever vanishes, then you'll be able to find more on the surface (there's an unlimited supply of red stars, so you don't have to hurry!). Collecting all 100 purple coins makes an actual Red Power Star appear. Collecting this will make the Red Star Item available in the Observatory for a special use.