So, I've launched my combat career by taking out ships sent to attack me when doing missions - doing OK so far, taken out about 4, but the last one I had no assistance from authority ships and it took AGES to destroy him even though he just seemed to sit there and let me hit him. I have a Cobra Mk III with 1G pulse lasers and 1D cannons, the lasers seemed to take out his shields pretty quickly but either I'm targetting the cannons wrongly or they're pretty weak - ran out of ammo before he went down. Any ideas?
If it was a bigger ship, your main problem could be in using only small weapons. Small weapons become less effective against bigger targets. Not because they have smaller overall damage output (They do, but not by that much) but because of the armour piercing capabilities. Your Cobra is capable of using class 2 (medium) weapons in the hardpoints in front of the cockpit, so I'd suggest using them to their full potential.
As you rightly observed, energy (thermal) weapons are more effective against shields, and kinetic weapons are great against hulls. As this armour piercing thing I was talking about apply to hulls and not shields, it is advisable to equip the kinetic weapons into your bigger hardpoints and energy weapons into your smaller ones. So the lasers in your class 1 hardpoints are a good choice.
But I would suggest several things for your class 2 hardpoints:
1) Equip a true class 2 kinetic weapons there. It will help kill things more quickly, once you strip down their shields
2) Cannons are great, but it's really hard to hit effectively with them - they are a bit on a sluggish side of projectile speeds. You can mitigate it by closing in on the target and hitting them from up close. But once you are close, you could as well use the Fragment cannons, instead. they are much better, but only at close range (They are basically a big shotguns)
If you want to stay in the safer distance, instead, I'd recommend trying out the Multicannons. They are much faster and track the target better (if you use gimbals) You will find out that although their damage per shot is low, they can overdose the target with lead pretty well.
3) Try targetting a specific module. In your left panel, there is a sub-targets tab (or you can bind keys/buttons to switch between the modules). If you destroy their thrusters, they stop. Destroying the powerplant renders them helpless and destroying the FSD prevents them from running and jumping away.
Multicannons in particular are really great for this, so check it out.
Good luck. o7