There are 2 types of cartridges, Passives and Action Cartridges. Passives happen passively on your character, while action cartridges require an input to use. Action Cartridges are always black, while passive cartridges are white, green, or blue.

Hexnuts are meta progression currency which allow you to purchase cosmetics, new engines, and Starting cartridges.
You can start with 3 passives and 1 action cartridge you need to spend hexnuts to unlock cartridge slots (locked slots will have a lock icon, the one in the image requires 150 hexnuts to unlock).

By playing the tutorial over and over again you will get 5 Hexnuts each time, to select the tutorial level click on the tool icon and then change biome to training facility.

The reason for this, is that connecting to a station will earn you 1 Hexnut. so 1 Hexnut for frist station, 1 Hexnut for 2nd, and 3 Hexnuts for beating the tutorial boss (biome junction). There are also 2 Cassettes that are tied to the Felix & Mathilde's Biggest Fan Achievement, 1 Cassette is in 2nd station if you pick the medium path in the first shop (top path), and 2nd Cassette is in the top right of the biome junction station if you break the trees. Check the jukebox in the main menu hub to see if you have got them.

You can also unlock cartridges for the notebook and in the cartridge hub by selecting a cartridge in the shop after beating the biome junction, make sure to go for new action cartridges first. You also get a hexnut for getting a new cartridge for the first time, and when you pick up a cassette for the first time.

Best action cartridges in my opinion are Jetpack and Throwing.