No Man's Sky is a game that encourages freeform exploration and experimentation, but the one clear objective you're given is to make it to the center of the universe. Here's how to make Warp Cells, which allow you to jump closer and closer to your goal.
Once you've constructed a Hyperdrive, you're tasked with creating a Warp Cell to fuel it. To do this, you have to combine a piece of Antimatter with 100 units of Thamium9. The former is easy, as you'll soon detect an Outpost with intelligent life. Go there and the Gek you find there will give you some Antimatter no questions asked. The Thamium9 is a bit more difficult, though, as it's several times rarer than Plutonium. Thamium9 is usually found on small plants or mushrooms that ping as an Isotope from scans, but the most efficient way to collect Thamium9 is by mining asteroids since it tends to be way more common in space.
After creating your first Warp Cell and jumping to a new system, you'll detect a Manufacturing Facility. Go to it and fix its malfunction to get the Antimatter recipe. To create your own Antimatter, you need to chain together multiple recipes to eventually end up with Antimatter. You start by crafting Suspension Fluid with 50 units of Carbon, which is fairly easy to procure from most plants you come across. Once the Suspension Fluid is done, combine it along with 100 units of Plutonium to create Electron Vapor. Chances are you'll have plenty of Plutonium on you at all times since it's the primary fuel source in the game, but if you don't, you can usually mine it from red-colored crystals. From there, combine the Electron Vapor with 50 units of Heridium and 20 units of Zinc to end up with Antimatter. Heridium can mostly be mined from the giant stone slabs that dot the landscape of just about every planet you land on. Zinc is a bit trickier, as they're found on plants that ping as Oxides. From there, you're another 100 Thamium9 away from another Warp Cell.
You also have the option to buy the individual ingredients from trade terminals and alien pilots. Trade terminals are most reliably found at Trading Posts and Space Stations, with the latter being much easier to find since they're in space with not much else to sift through. Skipping recipe steps and buying Electron Vapor or Suspension Fluid isn't advisable, as that can end up costing ten times as much as if you merely bought the minerals and worked your way up the crafting chain from the beginning. If a trader doesn't have what you need, wait around in a docking area for a ship with an alien pilot, as all of them have goods on offer. Buying the required minerals from traders is by far the easiest solution if you've got the cash.
Once you start gaining some Atlas Passes, building Warp Cells become a bit easier. Once you get the v1 pass, you can unlock canisters containing Suspension Fluid or Electron Vapor. Later passes will give you Antimatter and even complete Warp Cells occasionally.