There are two methods by which one can acquire a new ship, through financial distribution, or finding a crashed ship.
CRASHED SHIPS
Crashed ships can offer an immediate upgrade to a new traversal vehicle without spending money. It's a great way to expand your potential traversal ability, but only in certain cases. A crashed ship will be similar to your situation at the start of the game, stranded and searching for materials. If you happen across a ship that would be a substantial upgrade, don't simply switch to it immediately. Instead, look at all the materials you need to gather to fix the ship. If you're on a resource-barren planet, travel to one that will offer you more ease of collection, or buy the resources directly from the Galactic Market, If you're tremendously lucky, you'll be able to find the materials around the crash site itself. Simply gather what you need, and go into the ship comparison screen. TRANSFER ALL OF YOUR ITEMS to the new ship and make that ship your primary. Repair the broken elements, and you have a new ship.
BUY A SHIP
You can find other NPC travelers at space stations and outposts. Approach them and request to make an offer on their ship. If you have enough money, you can transfer your items to their ship and simply make it your own. Ships can cost anywhere between 250,000u and 5,000,000u.
SHIP TYPES
GEK: Gek ships have a focus on trading, and will usually have the largest inventory available.
VY'KEEN: These ships are more focused on combat, and traditionally come fitted with better weapons.
KORVAX: Considering their focus on scientific discovery, these ships will be more inclined towards travel and exploration.
Though the pre-release trailers and conferences spoke to the contrary, the ships all handle relatively the same, their only disparities being the focus of the life forms agenda.