How do I access my inventory?

Pretty much as soon as I was able to freely explore, my inventory was full. However, I haven't found anywhere in the menus or any button that will allow me to view my inventory. I can see my ships and pilots, the upgrade menus, and the mods, but nothing for my inventory.

How do I open my inventory? What can I do when it fills up?

Your inventory is broken into 3 types of items: organic, metal, and cores.

As you progress through the game, you will discover various types of bases (observatory, prospector, etc). Each of these types will accepts either organic or metal, and all will accepts cores. When you give them, in addition to the rewards below the item, the location will gain XP and eventually be upgraded (tracked by the circle next to the location name). Each location appear to have a max level of 3 (indicated by the 3 dots next to the progress circle).

So, if your inventory is full, you can go to one of these locations and sell off some things. All locations of a given type accept the same things, but hovering over the location on the map will tell you if it accepts organic or metal.