Gate Observatory
You'll start the game on the small island floating above the Comet Observatory but you won't be able to access it again until you've grabbed your sixth Grand Star by defeating Bowser Jr.'s Lava Reactor. You'll gain access to the Gate and the Garden Observatory simultaneously.
Gateway Galaxy
This is the very first galaxy that you'll explore and its mission serves as your tutorial.
Grand Star Rescue
On the first Metal Planet, you'll rescue a caged Luma by defeating a big Goomba to get its key. You need to talk to the Luma first. The Luma then turns into a Sling Star which you can use to get to a second Metal Planet where you need to fight another Big Goomba to free another Luma.
Follow the Luma to a Warp Pipe which will take you to an area filled with yellow Flipswitch Panels and moving electrical platforms. Step on every switch to shut down a machine in the middle of the area and grab your first Grand Star.
Gateway's Purple Coins
A Red Luma challenges Mario to collect 100 Purple Coins. There are 70 located on the planet and 30 in the air which can be reached using a new Red Star powerup that lets Mario fly.
Boo's Boneyard Galaxy
Race the Spooky Speedster by transforming into Boo Mario which will let you phase through walls and propel yourself forward with fans. Follow the arrow signs throughout the cave. If you win, you get the Power Star and can go back and pick up anything else you missed.
