I usually target only generators. Once they are gone, the ships/guns go silent. Lasers seem to work better for precise targeting.

This works for lower level POIs (like many Zirax ships), but I recently discovered some POIs have indestructible blocks that stop you from killing all the generators. Obviously in that case you do need to destroy the guns before you assault the POI on foot.

You're using 45% minigun and 45% laser turrets.
Miniguns have a damage penalty vs blocks and at the distances in space, a fair amount of bullet spread.
Laser turrets have a damage penalty vs turrets, and are best used for taking down shields.
Your weapon choice is well... bad. You need less miniguns and more rockets, plasma or even just cannons.

Do NOT target generators. There are multiple generators in enemy vessels, not just one slapped on after the fact. They will always be in armored locations so you will waste a sizable amount of ammunition trying to take out generators.
Go for the thrusters. Only the rear ones. If you take out all of a ship's rear thrusters, it will initate an emergency stop with the rest of the thrusters (needs those thrusters on the other sides to initiate a full stop) allowing you to focus on just the side you want to clear before hitting the cores.
Or, once stopped you can engage with an SV and board the ship and hunt the cores on foot.

Thanks. This is really helpful information.

I didn't realize miniguns had so much spread.

They don't have much spread, but when engaging a ship in space combat over 500m away when both you and the target are moving and your target is a comparatively small object, the chance of missing gets higher.

Makes sense.

So against destroyers, I assume we want a large fraction of missile turrets to destroy the thrusters?

To drop the shield, is there anything better than laser turrets? It seems like mounted weapons might be relevant here since there's no need to aim for specific elements when looking to drop the shield.

personally, I tend to rely on fixed weapons for the bulk of my dps. Plasma is fairly universal; decent against shields, and a bit of AOE to compensate for near misses. I'll have a mix of laser and canon turrets usually set to turrets themselves to soften the incoming fire, or to use as targeting tracers if I need to see where to aim. If you are planning on boarding to take it on foot, its good to have a single laser firing to keep the shields down in the mean time.

I know your question was about targets, so I'll throw in my 2 cents on that first:

For ground-based POI's I go for generators every time. Turrets yield useful resources and in my experience, most ground-based POI's are easily dispatched in this manner, while keeping the turrets for salvage. You can go in and core it on foot after (Please note that this is only for faction POI's, not the purple POI's).

For ships, I agree with Vermillion, taking out the ship's rear thrusters is quite easy and an immobile ship is easy prey and makes it much easier to deal with. Just beware that a lot of ships are still very lethal from the rear.

Now that's out of the way, weapon comp - I've tried lasers, missiles, railguns and plasma and I have to say, plasma seems the best of both worlds - accurate enough and good vs shields but still punch satisfying holes into things with some nice AOE damage (which is always better than single target). It works great on turrets or fixed. If you're only looking to core immobilised targets, then you might want to try railguns for your fixed weapons, but I prefer plasma myself.
Missiles are great, but a bit too devastating for my liking - I kill ships for fun and profit - not much left to salvage after a missile kill.

Some of the zirax ships have the shield generator in the nose behind red blocks, take it out with rails to keep shields from recharging.