Home Base is your starting point for navigating to each of the 4 categories.  Click on one of the 4 buttons in the middle of the screen to send Star & Comet to explore that category.
The 4 categories are  FUNDAMENTALS, GEO-SCAN, DISCOVER, & TIME WARP.  

Each time you successfully complete a category & the challenge that goes along with it, you will return to HOME BASE to receive your CRYSTAL REWARD.  You can keep track of which crystals that you have received by looking in the top center of the HOME BASE window.  Categories that you have successfully completed will have a DARK button with the category name written in WHITE.

Once you have received ALL 4 crystals (by completing all 4 categories and challenges), then you will return to your HOME PLANET to receive your promotion to Commander First Class!

Good Luck!!