If you have run out of inventory space, you aren't alone. No Man's Sky has hundreds of materials to gather and collect in lieu of the crafting system, and you spend a great deal of your time amassing items. Your inventory takes a beating as a result.
To upgrade your inventory, you will need to find an upgrade hologram. These are located on crashed spaceships on the surface of planets, usually angled toward the sky.
The difficult part is finding crashed spaceships. Due to their orientation, your best bet is to fly in low orbit around a planet searching carefully for them. Once you see one, land, and head inside.
In most crashed spaceships you will find a blue colored hologram that you can interact with. Interaction provides you with the option to upgrade your suit inventory.
Your spaceship also has an inventory, but it is locked by type. In order to gain a larger cargo hold, you will need to buy a different ship. Start saving!
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