[Part 1... The Basics]
The thread is 2 years old but the questions are probably still valid for new players. I had basically the same problems, so I am writing how I solved it and what bits of advice I found scattered around.
bificommander: And then we leave the solar system and get to the enemies with shields. And that's where my enjoyment ends. Now the battles become multi-minutes slogfest with all my ship's best anti-shield weapons, just to get a 10 second window in which I can hit the actual ship.
Well phrased.
The enjoyment does not need to end, though.
bificommander: And when you get to missions like the defense against the siege laser or the larger fleet battles, I lose all feeling of control and I get the impression that it's pure luck whether I manage to complete a mission or not.
That's a tough one. I am not really sure how I succeeded,, either, at the 3rd try. One key element is for sure to stop the siege laser from getting used by attacking one of the ships connected to get the siege running. And for this usually I had to leave the fortress shield and tried to get rid of the battleships as fast possible. No battleships no siege laser.
bificommander: Those shield-piercing precision weapons that I take with me in a vain hope to nail critical systems while the shield is still up don't do much.
There are 3 different weapon types:
1) Anti-shield (energy weapons): shells (normal), bombs (heavy, area)
2) Anti-device (tactical): light laser, heavy laser, gunboats (squadrons)
3) Anti-hull (destructive): guns (normal), plasma (heavy), torps (normal/heavy) and missiles (heavy, area)
This means there are 2 approaches possible vs. shields:
A) Tactical anti-device weapons
B) Anti-shield weapons
With only a few ships or even just 1 to your disposition, you should concentrate on one approach, either anti-shield OR anti-device weapons. You also need destructive weapons and often you haven't got enough weapon slots to use all 3 weapon-types. And it's not only a question of slots but also to achieve the necessary firepower to overwhelm recharging shields. With bigger fleets in your hands, you can, perhaps even should, use all 3 weapon types. Due to their different range, they work quite well together.
Before we go in depth into lasers and energy weapons some very important things about how some basics work in Nexus.
POWER SUPPLY, NUMBER OF WEAPONS, ATTACK ORDERS
My first thought to equip the ships in a way that each has a specialized role was not a good one. If this ship is lost you might be helpless because no other ship can take over the role. Another reason: power supply! Your ships, even when equipped with the best generators, can't use all their weapon slots at the same time. While they will do so when you force them by manually appointing each weapon to a target, their fire rate will be greatly diminished. It is better to let the inbuilt "battle computer" of the ship do its work:
* "F5 drain shields" to get your shell anti-shield weapons working (long range area energy boms are NOT used).
* "F6 attack device" for your lasers.
* "F7 destroy hull" for your destructive weapons.
* "F8 artillery": IMHO better stay away from this one but do it manually: missiles are always in short supply and area weapons need careful usage.
So if you see that none or not all of your weapons of a type do fire, this might be caused by
1) distance
The ship might be too far away and is still in the process to get into the range your weapons can fire.
2) position
Each device has a certain place at the ship and your ship needs first to manoeuvre into a position where it can reach and fire at the device. This is also true for your own weapons. For example a missile can't be fired if your ship shows its rear to the target, you need to at least slightly change the angle and turn back on course after the missile was fired.
3) power supply
As already hinted at above, mostly you have only enough power to use 3 weapons at the best recharge time at the same time and that's with a good generator. Might be even only 2 weapons if you have a weak generator. That's when you might manually raise the power to weapons (left lower corner) and what the energy cells in the equipment list are for, a power reserve which is then used up (shown in the pointed line above "Supply"). Or you save your energy reserve for other tasks and emergencies and cut down the number of this weapon type the next time you can change your equipment. Or get a better weapon generator.
4) weapon damaged
If you are under heavy fire, it might help to take a look at your repair list to see wether the weapon in question might be out of action. If that's the case, probably also other of your ship's devices are damaged and it is high time to set priorities for your automatic repair facicilites: there is a small number next to each listed device, higher numbers raise the priority the repair task gets. Once a device is completly down, it can be repaired only at a dock (next equipment phase). As nice as weapons are, higher priority should be given to your flaks, shield, engines, generators. Unfortunately your changed repair priorities get reset for each new mission.
[Part 2... lasers & gunboats vs shields]
A) TACTICAL ANTI-DEVICE WEAPONS vs shields
Device identification
The key for using tactical weapons is that you first need to identify the devices. When you click on an enemy ship a box pops up at the top of the display. At first you get only limited information about its equipment or none at all. The closer you get, the better your sensors (which should be switched on), the more information you get. Usually the first identified devices are the weapons in use. If you finally identify the shields, click on it, it starts blinking which is the commando for your lasers and gunboats to concentrate on that device. If there is flak, and you are using gunboats, priority should be given to shut down those first. Until the shield is identified, mark another device as target. What device is perhaps a matter of taste. My personal priority:
1) combat engine: That engine is specifically to make things harder for gunboats and lasers, it rolls the ship so the targeted devices get out of the fire line. Egnines are also easy targets and bringing it down will go quite fast (if you are somehow in position or have gunboats).
2) flaks: Only when you have squadrons out there. Your fighters take care of enemy fighters, your gunboats of the flaks and afterwards your squadrons are almsot totally safe.
3) missile and energy bombs: Only when some of your ships are endangered by them: not in close combat but at a distance and might get targeted.
4) shields
5) weapon generator or other engines
6) anti-shield weapons
7) other weapons
Gunboats
As a matter of fact Gunboats also belong to the tactical anti-device weapons and they are quite good at it when it goes against single ships or a small group. Against bigger groups with lots of flak systems they tend to die like flies, though. With ctrl-leftclick choose the gunboats and point them to one ship you want the devices disabled asap. If there are enemy fighters/gunboats, send out your own fighters for defense, mostly the easiest order is also the best: with "F8: patrol fleet" they will take care of the enemy squadrons and will also try to shoot down incoming missiles. You have already noticed that the short cuts for orders are different for ships and squadrons, haven't you? F8 for ships: artillery, F8 for squadrons: patrol. Sometimes, if the enemy uses missiles, it is wise to deploy some fighter squadrons to specifically defend your biggest ship (choose the squadron and just click on the ship they shall defend). Your squadrons are also faster than your ships, so in an ideal situation your squadrons have already disabled the shield when your ships come into range to do the rest.
As long as 1 ship per squadron survives, the squadron will be automatically and without cost brought up to full strength between missions. In long battles it is worthwhile to call home to ship (F1) heavily damaged squadrons where they get repaired and refitted. The state of your squadrons is shown with the small bar under them and more detailed in a box located at the lower left corner when you click on a squdron. By the way, the numbers there show how many shots they still have, usually plenty, for bombers number of bombs are shown.
While gunboats are fine (and fun), priority should be given to your ship weaponry, though.
Lasers
They come as light and heavy ones. Since the shield is still up, the light lasers aren't of much use IMHO, so you need heavy lasers which have the additional benefit of better range, around 10k but slower recharge. Between 2-3 heavy lasers, depending on ship class and power supply, you should equip per ship if you go the laser way vs shields. After we have covered the basics already above, the appproach is simple and streaight forward. Choose your target-ship wisely, make sure to target the shield by clicking on it, choose several of your own ships with ctrl-leftclick and order them with F6:attack device to do the job as fast as possible. Once the shield is down, switch to F7:attack hull.