The star map is how you'll navigate to different star systems. In this screen I have a lot of choices for where to travel within my range (green ring). But sometimes the stars are few and far in between, which means you'll need to backtrack, resulting in almost certain death.
A little later in the game you'll find different ships that can hold more inventory. These are great to have because you can stock up resources to keep your ship moving along with metals you can use to craft technologies.
Touch the wrench-and-hammer icon (upper left) to open the Technologies screen. Here you'll be able to see the tech you already have as well as new tech you find along the way. With the right metals, you'll be able to craft new modules for your ship.