You begin on a random planet in , in a galaxy full of more planets than your mind can comprehend and with nothing more than a wrecked ship, an empty exosuit and a puny multitool.
This inaugural ship comes with a meager 15 storage slots. Four of those slots are taken by the required bits and bobs that make the ship fly and survive between the harsh planets -- the Launch Thruster and Pulse Engine, both of which are heavily damaged, a Deflector Shield and the Photon Cannon. There's also a Carite Sheet, which you will eventually use to repair some of the damages.
If you preordered the game, you're granted the option to replace the first ship with the bonus DLC Horizon Omega ship, which has one additional slot and a Hyperdrive (for warp speed travel) preinstalled. If you choose this ship, you still need to make the repairs to get it off the ground.
Once you get the ship in the air, you'll soon find that the 15 or 16 inventory slots you start with on your first starship simply aren't enough. You will need to expand and upgrade to make your journey more enjoyable.
Upgrading your ship
Upgrading is the name of the game in No Man's Sky, and what you're after is more storage space. Your inventory space is crucial for mining and storing valuable resources or holding on to important things like Atlas Stones. These slots are also used for installing new technologies that make your ship more efficient, such as by making your Hyperdrive travel farther, cooling the Phase Beam so it shoots more than 1 second without overheating or improving shields for combat.
You will find new technology to install along your journey, but without upgrading to a new ship, you won't have room for the add-ons and all the other things you need to carry with you.
The most direct way to upgrade your ship is to interact with other travelers at space stations, trading posts and outposts throughout the game. Once another ship lands, approach it and start a conversation with the owner of the ship. You will be given the option to make an offer on that ship, which varies wildly in price. The first ship I purchased from another space traveler cost me approximately 700,000 units for an additional slot. The second ship I purchased cost me over 3.4 million units for a total of 24 storage slots.
The maximum number of slots you can get in a ship in No Man's Sky is 48 and will cost you upward of 100 million units.
Doing the math and comparing that with how slowly you earn units for mining valuable resources (and considering the fact that your existing ship apparently has no value after you purchase it), it would take ages to go about maxing out a ship in No Man's Sky solely by trading up.
But there's a better way.
Look for abandoned ships
If you've wandered around on several planets, by now, you've likely stumbled upon a crash site or two. At a crash site, if you approach an abandoned ship -- which will have either one more or one fewer storage slot than the one you're currently flying -- you have the option to transfer all your loot over to the new ship, claim it as yours and leave behind your current one.
The problem is, you're going to have to repair the abandoned ship just like you did in the beginning of the game. And depending on the climate of the planet you're on, the hostility of sentinels and how plentiful the flora and other resources on the planet are, it may be a wise decision to leave the crashed ship be.
That said, crash sites aren't all that uncommon, if you know how to locate them.
How to find crash sites
First, search for a Signal Scanner, which you can typically find by cruising around in your ship above the surface of a planet. Signal Scanners are small, trapezoidal boxes that emit an orange beam out of the top, well into the air. (If you can't find one during the day, particularly on a mostly red, orange or pink planet, they're very easy to spot at nighttime.)
Once you find one, make your way to it and craft a few Bypass Chips in your exosuit inventory using 10 Plutonium and 10 Iron. Walk up to the Signal Scanner, use one Bypass Chip to access it, then select Transmission from the options. It may take a few tries, but the Signal Scanner should eventually show you the location of a Tranmission Tower. (If you know what a Transmission Tower looks like, you can skip the Signal Scanner altogether.)
Make your way to the Transmission Tower, which will be a building with a large antenna sticking out of the top. Inside, you will find a terminal. Interact with it and solve the problem to decode the distress signal. If you solve it, you will be given the coordinates of a crash site.
Once you arrive at the crash site, approach the abandoned ship. To initiate the transfer to the new ship, press the same button you normally press to enter your ship (by default, it's E on PC or Square on PS4). Check the number of slots the crashed ship has. If it doesn't have as much storage as your current ship, it's probably best to leave it...unless, of course, it looks awesome.
Salvage tech from unused crash sites
However, you can scrap the technology on the abandoned ship and use it to help repair the next one you come across. To do this, switch to the crashed ship, but leave all your resources in your existing ship. Then open the inventory of the crashed ship and dismantle all the add-on technologies (middle mouse button on PC or R3 on PS4). This will destroy the technology and recover some of the resources that were used to create it, often scarce materials. Once you've deconstructed everything, go back to your original ship and switch back to it, taking all the important goods with you.
When switching to a crashed ship that has more storage than your current one, you can use this same tactic to scrap all the technology on your ship to help repair the new (crashed) one.
Frankly, this is a grueling process. It's a grind to get a fully maxed-out ship. But by comparison, repairing crashed ships is a much faster way to increase your inventory space than purchasing ships, which requires you to spend hours upon hours of harvesting resources to sell, just to earn enough units to purchase one upgraded ship.
In that same time, you could easily find and repair multiple abandoned ships and increase your inventory space without spending a single unit.