No Man's Sky puts the sky in the palm of your hands once you gain the ability to pilot a starship. Here's how to expertly pilot your starship.
Once you're in the air, you begin to fly automatically without needing to press any buttons. Steering is tied to the same controls as regular movement, and they're not inverted by default. You can also roll left or right, but unless you're trying to get through a narrow space, it doesn't accomplish a lot besides doing a barrel roll to grant you a tiny bit of evasion in a dogfight. While you're in the air, you're free to open your inventory without worrying about falling out of the sky. You're eventually going to want to apply some thrust, though, because it would take forever to get anywhere without it. This speed is generally recommended as the accelerant of choice during firefights with other ships, as you can keep moving fast without sacrificing maneuverability.
Holding down the Boost button will cause your ship to lurch forward, more that doubling your speed. However, you won't be able to change direction very quickly, so Boosting isn't advised in the middle of a fight unless you're intent on retreating. You're mainly going to want to Boost while moving from location to location on planets, as they can often be upwards of 20 minutes away without it.
If Boosting is still too slow, you can utilize the Pulse Drive in space by pressing both Roll buttons at the same time for five seconds, putting you into a sort of sub-light speed state for a short amount of time. Be careful not to overshoot your destination, though, because it doesn't take long for you to be flung far off course. Your best bet for using the Pulse Drive is to aim at the destination marker of the place you want to go before you launch. Be careful if the destination arrival time is under an hour, though, or you risk overshooting. You can never use the Pulse Drive planetside, and the pulse drive will automatically stop once you enter a planet's atmosphere, fly near a space station, or fly into a group of pirates.
Flying your starship will expend fuel, but even the most basic engines are pretty fuel-efficient, requiring only a modest fill-up of Thamium9 every once in a while. Should you ever run out, you're stuck with Boost as the fastest speed you can go, but shooting nearby asteroids should score you enough Thamium9 to fill you up. The default Launch Thruster can guzzle through around 25% of its Plutonium stores with every takeoff. Make sure to always have plenty of Plutonium so you can keep taking off, but if you're hurting for Plutonium, start walking to surrounding locations before taking off and moving on.