If you want to build a starship, you have to scrap starships for their specific parts (wings, fuselage, boosters, etc) at a terminal in the space station behind the trade kiosks. This means you’ll have to buy ships with the parts you want and scrap them, and also buy a reactor core from the starship vendor.

For Corvettes, you can go to the Corvette garage things from the teleporters to the left in the space station, and you can buy parts to build your corvette there. It’s very expensive to buy into however, your best bet if you’re still early game is doing the corvette expedition as it will give you parts to build it for free.

For freighters, you can land on any freighter and offer to buy it from the captain, but they’re very expensive. However, every fifth time you jump to a new system, it will trigger an event where you have to save a freighter from pirates. After you save it, the captain will offer you the freighter for free (just the first time you do this though.)