How to dogfight in Elite: Dangerous?

The only thing that will let you win dogfights consistently is a lot of practice, however you mention that you are not able to fire back at the enemy while escaping and there is a trick to do that.

Firing while escaping

Turreted weapons

There are essentially two ways to fire back at your enemy while trying to create some distance.
The first one is pretty straightforward: Turrets!
If you take turreted weapons instead of fixed or gimballed ones they will be able to fire at the enemy while you are flying into the opposite direction. This however will cost more money and they will perform worse than a fixed/gimballed weapon if you can just fire at the target. This brings me to the second point (and I wish someone told me that earlier):

Utilizing flight assist while escaping

This is probably the single most important thing that makes dogfights a lot easier if mastered. If you have an enemy right behind you you can boost away from him, turn your flight assist off and pull your ship up (or down) so you look back at the enemy.

Turning off flight assist will let you pitch while continuing to fly into the direction you were originally facing. Additionally when boosting your pitch speed is always as if you were in optimal flying speed.

This method will take some getting used to but it helps a ton when being chased by more agile ships.

General information

Targeting enemy systems

Another thing that will increase your combat efficiency is targeting enemy sub-systems. Don't just lock onto a target and start firing. Try to take out specific systems of the enemy ship. If your enemy is escaping try to shoot his boosters this can cause them to either malfunction or you can destroy them completely which makes the target a sitting duck.

Targeting enemy weapons will limit their offensive capabilities and targeting their power-plant can shut down an enemy ship completely. Know what systems are important in a fight and try to disable them.

Key bindings/controls

It's extremely important that you're using controls you're comfortable with. Don't be afraid to reassign keybindings if it makes it easier for you to maneuver your ship. Combat is the only thing in Elite Dangerous where speed and precision matters when controlling your ship. For me I've changed the targeting on the sub-systems of a ship to mouse-wheel up and down since it makes I find it a lot easier to cycle through them when compared to pressing a specific button in order to do so. Be creative and try things out and find out what's best for you. Also this applies to KB+M, Joysticks and Controllers.

Choose the right equipment

Your currently mentioned weapon setup (equal amount of multi-cannons and lasers) is pretty safe and works somewhat well against everything but it doesn't really excel at anything. If you know you're going up against mostly slower and bigger ships that are easy to hit consider bringing a rail-gun with you. Against unshielded targets maybe missiles will easily secure you some kills.

Also Take the right ship. You don't want to go kill-farming in conflict zones in a Type-10. An Anaconda will be the better choice if you're looking for a heavier ship in that case.

Practicing dogfights

If you're not too confident in your fighting abilities yet I'd highly suggest working your way up from beacons, to resource extraction sites until you finally will be able to participate in conflict zones.

Conflict zones are extremely hectic and it's common to be outnumbered five to one which is a difficult fight even for more experienced commanders. If you want to straight up dive into conflict zones you can do that but try to single out enemies so you can practice the fighting. Also don't be afraid to take on bigger ships or higher rank NPC commanders.

Combat Zones in Elite are considered an endgame activity; especially so since about 2 years ago they allowed higher-level NPCs to have engineered modules, and their AI has been significantly improved. Military-type ships - that is, the ones you usually face in a Combat Zone - have also been equipping more heavily reinforced armor compared to the past, making them tankier.

You did not mention engineering in your post, so I'm assuming your ship is not engineered; while time-consuming, engineering will improve your ship's performance manifolds: your damage output will go up roughly 50-60% (depending on weapons and blueprints), and your ability to withstand damage will more than double.

Having said that, there are a number of things you can do, even in a small ship, to improve your odds.

Throttle Management

Your turning speed is affected by your throttle position (with Flight Assist On) and your pips to ENG (with Flight Assist either On or Off). Keep your speed in the blue zone to improve your handling. Engineering your Thrusters will also help, as does boosting. Since turn rate is increased in Flight Assist (FA) Off, many players switch to it, boost, turn, then switch back to FA On.

Of course using FA Off permanently will greatly increase your time-on-target, but it does have a steep learning curve.

Aggro Management

NPC pilots in a Combat Zone (CZ) will tend to prioritise players, however you can mitigate this in several ways:

• Attack ships on the edge of the furball, not right at the center;
• Attack ships that are already being attacked by allied ships, possibly bigger than yours;
• Take pauses from time to time to recharge shields and pips; enemy aggro is based on damage dealt, so keeping a low profile for a while will make it less likely for nearby enemy ships to switch on you.

You also don't need to be at max speed all the time, especially if nobody is firing at you or your current target is attacking someone else; just slow down and match your speed to it.

Still, sometimes things may go wrong and it may be time to leg it; when that happens, fire a Chaff (because you have Chaffs, don't you?) and run away while rolling and strafing in either 4-2-0 or 2-4-0 configuration, depending on the speed of your pursuers. The small size of your ship will help avoid shots.

Pips Management

Newer players may not be aware, but having more pips to SYS does more than increasing the recharge rate of your capacitor, it actually makes your shields more resistant to all damage sources (including Absolute damage); the increase is slightly exponential, but peaks at a massive 60% when having 4. That is, an attack that normally deals 100 damage will only deal 40 damage if you have 4 pips to SYS at the moment when the attack hits.

Conversely, turn speed is affected by the amount of pips you have in ENG.

It is very helpful to change pips configuration on the fly as needed, I use a program called AutoHotKey to write pip-management macros.

Find a partner

It may not be easy to find one, but if you team up with another human player you won't have to share your earnings: if you are in a wing you both will get full credits for every target destroyed, assuming you both dealt damage to it recently.

Final Remarks

Having a large ship can help; they take more punishment, they can equip Ship-Launched Fighters that are great for distraction, they have plenty of internal slots for Shield Cell Banks and additional armor, and they can spend all day in 4-0-2 for maximum lazyness (if they constantly reverse-turn to keep their aim on target). This is how I started, even before engineering became available. You can consider saving up credits to buy one, or alternatively start engineering your ship - or both, as a fully-engineered large ship is effectively unstoppable in a Combat Zone.

Source: I'm a Flight Instructor for Spear, a PvP group comprising over 12 different Squadrons.