Aqcuire Antimatter Recipe
In the first Star System you start in, a random alien will give you an Antimatter up front to craft a Warp Cell to fuel your Hyperdrive as part of the tutorial - but once you get to the next Star Syatsm, you will need more Antimatter to progress. Luckily, as you start exploring the new system, you'll get a notification to visit a random shelter on a nearby planet.

Head down and investigate to run into yet another alien who will finally give you the recipe to make your own Antimatter.

Craft Antimatter
To make your own Antimatter, you'll need 1 Electron Vapor, 50 Heridium, and 20 Zinc. This is where things get interesting, as the Electron Vapor is another item that you can either buy off the Galactic Market - or craft from the recipe you should already have available. However, to craft the Electron Vapor, you need 100 Plutonium and yet another technology component - Suspension Fluid. This item can also be bought from the Galactic Market - or you can locate its crafting recipe. You can sometimes find recipes for technology components by exploring planets and looking for manufacturing plants or depots, and fixing the terminals inside.

Get the Plutonium from red supply boxes around shelters and the red crystals found in most planets - usually in and around caves. With the Electron Vapor made, you can combine it with Heridium found in large rock pillars with blue cracks, and with the Zinc found by interacting with certain yellow plants.

Craft Warp Cells
Now that you know the recipes to make Warp Cells, Antimatter, and Electron Vapor (and hopefully at some point Suspension Fluid), you'll be all set to craft as much fuel for your Hyperdrive as you want, and start jumping between Star Systems. The more fuel you make, the farther you can jump - which can be compounded by either finding and installing the blueprint for a Warp Reactor - or finding/buying a Starship with one already installed.