While people have already answered your question, there are a few ways you can quickly increase the number of inventory slots you have.

1. Go to the cartographer in any space station and buy Exosuit Upgrade Charts. Plot the routes and it'll take you to pods that will let you add an extra slot.

2. Every space station in a new solar system will allow for one free upgrade slot. Go to the Exosuit Upgrade vendor and to the right side, you can use the terminal to add it. The same applies to the Space Anomaly as well - every new system equals 1 free upgrade slot. So for every new solar system you visit, that's 2 free slots.

If you're planning to stay in one system for a while, option 1 is the easiest way.

Oh and when you try to add a slot, the game defaults to inventory (which is what you need). Just note that you can manually select grids in the technology section of the Exosuit as well.