For aiming:
You may notice a small red cross that appears when you highlight an enemy. Always move your cursor to highlight this, rather than highlight the body of their craft only. You have to account for the delay from laser fire to actually travel the distance and hit the target. That's why you should aim at the small red X that appears infront of their travel path, as this will put your laser fire into their oncoming path.
Also move your cursor as quickly as possible to their red arrow if they attempt a fly-by.
Lock onto their craft properly:
Take the time to hit space bar and autopilot. Quickly highlight their craft in the bottom left of the HUD and keep clicking to highlight the part of their ship you wanna target. This helps focus your attack. Then switch back to free-fly mode and chase them down!
Use Afterburn:
Use the Afterburn (shift key) to catch them up and keep up with them. Use it sparingly because you'll need the gas to evade other fighters trying to blow you to hell. It's a valuble tool to help you maneuvre. Look closely at their movements. They speed away, and fly towards you, speed away and fly towards you. Use the afterburner to your advantage.
Get outta there!
Finished the job? Taken out your target? Why hang about? Hit the spacebar key and go into autopilot for a second. choose the "go to" icon for you next waypoint. This disables your weapons but makes you move at speed to get out of a tight situation. Hit the space bar again to assume control (while staying in the fast speed mode). Keep strafing left, right, up and down and get clear of any danger. They'll find it hard to catch you up.
Hope that helps.
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see you, space cowboy...