ACQUIRE HYPERDRIVE TECHNOLOGY
Soon after making it to space, you'll pick up a Beacon signal. Follow it to a new planetary location, using your Pulse Drive to shorten the trip. Land and enter the Outpost. Once you land, enter the station and talk with the alien there. He'll give you a Hyperdrive Blueprint

BUILD A HYPERDRIVE
To build the Hyperdrive from the blueprint, you're going to need two ingredients - A Dynamic Resonator and 200 Heridium.

For the Heiridum, you'll have to go through the same process you did to get off the first planet - namely, finding a vein of Heridium that's hopefuly on the planet you are on: Look for giant rock pillars with blue cracks in it.

As for the Dynamic Resonator, unless you've investigated enough alien factories to find the recipe for it, you'll have to buy it off the Galactic Market. To do so, gather some significant amounts of whatever precious elements are found on your planet - like Emeril and other minerals, then travel back to space and visit the Space Station. Purchase a Dynamic Resonator onboard at the market terminal, selling your precious gems and commodities to help pay for it. You should now have the ingredients needed to make your Hyperdrive.

FUEL YOUR HYPERDRIVE
To fuel your Hyperdrive, you need a Warp Cell - but things aren'y that easy. You'll have the recipe for the Warp Cell already, which requires 100 Thamium9 and 1 Antimatter - which you don't have the recipe for.

When you take off again a new signal should appear. Follow the signal to a facility and talk to the Alien inside, where he will give you Antimatter. Now all you need is Thamium9, which you can easily find by destroying small asteroids, or inspect and interact with red isotope plants on most planets. With both ingredients, craft your Warp Cell, and use it to fuel the Hyperdrive, which should put it at about 20% fuel - enough for you to warp to the next star system!