In the manual, they give you these checklists for each and every possible procedure; firing gun from static mount, firing gun while tracking, firing gun from static with ranging, etc.
Imho a better way is to learn the different functions of the system, and mix and match what you need on the fly. I think knowing the system makes it easier to use it, as compared to remembering 15 different checklists.
Basic thing is to enable "master arm", nothing will work without it.
To bring up targeting reticle for gun or rockets (static fire), simply select the hardpoint (rockets) or enable cannon.
Note that the cannon overrides other weapons: you can switch to cannon with any other other weapon selected, but you'll need to disable it again before you can fire anything else.
The problem here is you're "firing from the hip": the reticle is only accurate for a particular range, so you'll need to eyeball the actual range and estimate proper leed/elevation. Need experience (or some spare ammo) to get this right.
A slightly more advanced method is to include ranging; the only extra tool you need for this is the laser, and set the Shkval mode to "GS" (switch next to laser) . Enable the laser, maneuver to place the reticle on the target and push the "lock target" button. A countdown of a few seconds will appear in the HUD, after which the reticle will jump. It now displays a proper impact point for the selected weapon, based on the range that you entered. Grants you much better accuracy and takes out the guesswork from "firing from the hip", though you should still leed moving targets. Note though that if you're moving at high speed, the range will have shortened in those few seconds.
You'll get the highest degree of accuracy with auto-tracking. It is also the most complex. Works for the cannon and Vikhrs.
In addition to the laser, you need to operate the Shkval. First, press the "uncage" button. The display comes alive and a reticle pops up on the HUD to indicate where it's pointing. Now pan it over your target and press the lock button. The bracket should start to jitter a bit, indicating the system is actively analyzing and tracking. If you have the laser active, a range will display at the bottom as well (if not, activate laser and press "target lock" again. You might realise now that the "lock" button doubles as rangefinder button).
If you have the cannon active, you'll notice a large bracket displaying in the HUD. It indicates the maximum area the gun can traverse. Make sure the target is within this limit before opening fire.
With Vikhrs selected, a reticle will display on the HUD, which will drop down to the same elevation as the target; the launcher is automatically depressed to line up the shot, but it can't swing side to side, so you'll need to turn the helicopter to line up the reticle with the Shkval reticle before firing.
In all modes, you'll notice the "firing authorised" indicator in the HUD and Shkval display; it looks like a "C". Without this, weapons will not fire. Firing is authorised when conditions are met; target is not too close, not too far and within launch window. You can override this by flipping a switch, allowing you to fire a weapon whenever you wish (American style).
Set to "MAN".
There are also 4 modes you can use to improve targeting and make your life a little easier.
"AUTO TURN" makes the helo turn to where the Shkval is pointing, basically lining up shots with Vikhr automatically, though you might need a little rudder input to counter wind.
"MOV GND TGT" should be enabled when targeting moving vehicles. As far as I know, it only tells the cannon to lead the target, though it might have some impact on Vikhr too.
"A/A" should be used against airtargets. It sets a proximity fuse on the Vikhr, increasing the odds of damaging a target when the missile misses direct impact.
"A/A H O" is basically the same as the previous, but the fuse is set to go off earlier. This should be used for airtargets that are coming at you.
Some other things to note on this panel: the radio selector knob should be set to "GRND CREW" if you wish to refuel or rearm. You can just open the cockpit door and shout when the engines/APU are not running, but otherwise you'll have to use the radio (make sure it's on, too).
The "HMS" switch controls the helmet mounted sight. Flip it, and a reticle flips down in sight. You can turn your head to look around and line up targets. When lined up, press the "uncage" button, and the Shkval will uncage and automatically slew to wherever you were looking. Can be used to quickly line up targets, especially with "AUTO TURN" engaged (beats the painfully slow panning in most cases).
The "AT" / "GS" toggle is used to control the Shkval mode. "AT" means auto-tracking, which is the default mode. "GS" means gunsight, and it activates, slews and locks the display to the reticle displayed in the HUD. You can use it to get a close up view of what you're engaging with rockets or cannon, but imho it doesn't offer an improvement in accuracy over the reticle in the HUD, and the field of view is so narrow that it is rather cumbersome to target with. Good for ID-ing targets though, but if you're close enough to deploy cannon or rockets effectively, you probably made up your mind about whether or not it should be killed.
Note that it must be set to "GS" to use rangefinding when firing without auto-tracking (locking the Shkval on target).
There's a function called "targeting mode reset". It's used to reset the weapons system, caging the Shkval, disabling any selected modes and deselecting hardpoints. It returns the HUD to default mode, which is much better when you're moving about (if the HUD does not return to normal after using this, you have the cannon active; as stated before, cannon overrides anything else, including the reset command). Default mapping is
backspace.