In No Man's Sky, you're marrooned with nothing but a ship and your smarts, forced to cobble together upgrades that will eventually lead you to the center of the galaxy. One of the most vital is antimatter, a necessary component for the hyperdrive that allows for long-distance travel between star systems.
Like many of the upgrades in No Man's Sky, the specific path toward your goal may vary. Since the game is procedurally generated, it spreads signposts for specific pieces of upgrades on a wide variety of planets, which may be more or less hostile to you than others. Part of this process will definitely involve a fight, so it's best to come prepared with some weapon upgrades, like the Boltcaster upgrade for your mining tool.
Without antimatter, you're barely getting out of a star system. You need it to build a Warp Cell, which fuels the Hyperdrive. You can't mine antimatter, and you can't buy warp cells as standalone items. That means at first, you'll have to forage for your own.
The easiest method of obtaining antimatter is to head to the space station where you found your Dynamic Resonator, and simply purchase it. Like the Resonator itself, it's likely to cost a pretty penny. Come equipped with some high-value resources like precious metals to sell, or just hop a few planets making discoveries until you fill up your coffers with units.
Alternately, you may be able to find antimatter on one of the nearby planets, like the one where you found the hyperdrive recipe. That will take more scavanging and time, so it's up to you to determine if you'd rather go the quick and expensive route, or slow and thrifty.
Once you've obtained antimatter for the first time and traveled to the next star system, you'll get a cryptic message about a manufacturing facility. Go check it out, and be prepared for a fight. The facility will be locked with a steel door that needs to be destroyed. However, attacking the door will awaken the nearby Sentinel robots, who will attack you and call for even more deadly backup.
Grenades are your best bet, since they do lots of explosive damage very quickly. Failiing that, use your Boltcaster to chip away at its health, while occasionally looking for cover to reload your weapon. It will probably take about 4-5 passes with a full clip of your Boltcutter, so practice reloading it with energy through the menus quickly. Don't worry about engaging the Sentinels themselves. They'll just send more, and once you're through the door they back off anyway, so your goal is to get in as fast as possible.
Once you're inside, you'll find the antimatter recipe sitting on a control console. That will allow you to build antimatter, which in turn allows you to build warp cells.