The next goal for any new space explorer is to build the hyperdrive, the device that allows travel between systems.
The first step on this journey is to follow a beacon that will be activated upon entering space. This beacon leads to a nearby planet, which can be reached using your now recharged pulse drive. Upon entering the planet and reaching the beacon, you will find a building with an alien inside. The alien will be speaking in their native tongue, and you will be unable to converse with them. They are quite generous, though, as they will give you the blueprint to build your very own hyperdrive.
Next, you must actually build the hyperdrive. This is a two ingredient blueprint, the first is a Dynamic Resonator, and the second is 200 units of heridium. As I mentioned in the last entry, heridium can easily be mined from the large blue deposits, and a single one should be enough for the build. As for the Dynamic Resonator, you have two options: Buy it from the Galactic Market, or continually visit outposts and shelters in this system until you've acquired the recipe for it. The choice is yours, but either one will require exploration and resource gathering. Once you have both requirements, build the hyperdrive in your inventory,
Now, you must acquire fuel for the hyperdrive, of which there is only one option. You must build a Warp Cell (which you already have a blueprint for), which requires Thamium9 and antimatter (which you don't have the blueprint for.) Leave this planet and enter space once again, destroying asteroids to preemptively build up Thamium9. Upon re-entry to space, a new beacon should appear on another planet. Follow that beacon, just as you did the first time.
At the beacon, you'll find another alien, speaking the same foreign language. After a short, unintelligible conversation, he will present you with antimatter. Generous. Craft the Warp Cell using the antimatter and the Thamium9 you received from destroying the asteroids like I know you did, and you have yourself a pristine Warp Cell. Use it to fuel your hyperdrive to 20%, just enough to reach the next star system.