Unlike in Frontier or First Encounters, there is no autopilot in Elite Dangerous! There are some in-flight aids, but flying between locations will require manual flight.
The “preferred” way of controlling ships in Elite Dangerous is pitch (aka up and down) and roll (aka “spin”) - apparently Frontier have deliberately made yaw (aka left and right) deliberately slow as they want to promote pitch and roll as the preferred way of controlling the ships.
The rationale for this was that Frontier were concerned that by having a fast yaw and pitch would cause multiplayer games to become simple matters of stopping the ship and then just rotating around to point-and-shoot at the target (the so-called “turrets in space” argument), rather than the “dog fighting” that is required by the deliberately ham-strung yaw rates.
This can be counter-intuitive if you are used to FPS style games where left on the mouse means “turn left” in the game (in Elite Dangerous it would actually roll the ship, spinning the screen). If it helps, try to think of this as how planes fly - to go left, they roll left then pitch up - they don't make the entire plane turn left using the rudder controls.
How well your ship rolls, pitches or yaws is dependent on the speed you're travelling at. There is a blue "sweet spot" on the speed meter to the right of the scanner that shows the optimal turning speed for your craft - if you go outside of this range your ship will be more sluggish.
If you adjust the power allocation to the engines, this sweet spot will move appropriately. For example if you add more power to the engines, the sweet spot for turning will be at higher speeds.
Even if you are in the turning “sweet spot” sometimes it is still hard to pitch quickly enough to make your ship face a target ship, often it is the case that you eventually end up chasing each other around in circles.
A good tip to get around this problem is to bind a key to 50% or 100% reverse throttle, then as you pitch to try and face the target, reverse your throttle so you are going backwards whilst still pitching. Once you have the target back in front of you you can then either return to forwards throttle, or continue in reverse if you like.
This has the effect of adding extra space between you and your opponent by breaking out of the turn pattern you are in, reducing the angle that your ship needs to cover when pitching and making it easier to get the target back in front of you.
In space there is no friction or drag to slow your ship down and this is the same in Elite Dangerous. If you command your ship to fire its roll thrusters to roll left, it will continue rolling left as there is nothing to stop it (no friction, no counter-thrust etc).
Flight assist is a feature available in all Elite Dangerous ships (so far) that automatically compensates your ship's thrusters with counter-thrust make flight more controllable and intuitive by counter-acting your inputs to stop your ship's movements automatically.
For example if you input a roll to the left, flight assist will counter-act that initial input with a roll to the right so that the ship stops rolling automatically.
If flight assist is disabled it will not counter-act the roll, and your ship would continue rolling.