When you first manage to lift off into space in No Man's Sky, you can travel between planets in your home star system, but you can't escape it until you build a hyperdrive. Here are the steps you must take to build the drive and truly break away from home.
Due to the procedural nature of No Man's Sky, each person's building experience may vary a bit. The general gist of the quest seems to be the same across all games, though.
-When you first escape the atmosphere of your home planet, you'll get a message about a nearby beacon. Travel to the planet or moon the beacon is on, and enter its atmosphere.
-Zoom in on the beacon's location. Don't overdo things with your boost, or you may zoom right past it. Alternate between boosting and drifting until you reach your destination.
-When you're close to your target, land. Walk the rest of the way if necessary, and keep an eye on your life support and thermal protection if applicable.
-At the marked beacon, you should meet a Gek (one of the lizard-like aliens that populate the universe). He'll give you the plans for a hyperdrive.
-You'll need a Dynamic Resonator for the drive as well as some common isotopes. Travel to the Space Station in your star system and buy one from the merchant terminal or from one of the merchant Geks.
-Once you assemble the drive in your starship's inventory, you'll need to fuel it. Doing so requires a warp cell, which, in turn, requires antimatter.
-If you're lucky, one of the Gek in the space station may be selling antimatter. Otherwise, take off again. You should receive another message from one of your system's planets. Again, zoom in on its source and talk to the Gek who signalled you. He should give you antimatter for your troubles, possibly after asking for Carbon in return.
-Now your hyperdrive is fueled up enough to make your first jump from your home star system. Happy adventuring, cosmic explorer!