Bananas,
Okay, here goes…
For starters, the sheer amount of weapons and such on your ship makes things kind of confusing all on its own. Basically, I can walk you through the UI, but I wouldn't want to try actually fighting a battle with this ship. In addition, some parts of this, like determining range to target, require you to muck around in other parts of the fleet window. I've asked that range information be included in the list of potential targets if possible.
Here's how things work.
You have to drag weapons to the fire control you want them to be assigned to, or just use the auto-assign FCs, button. If you check the "Assign all," checkbox, that will automatically assign all weapons of the same type to the same fire control. There's also a variety of fleet buttons which basically copy the FC assignments or targeting info from your ship to others in the same fleet, of the same class.
You then have to drag the target from "Potential targets," to the fire control you want to use. All this is indicated in the tree view as you do it. Finally, you can click "Open Fire all," or "Open fire BFC," or whatever.
Note a few things.
Your cruisers do not have missiles. You need to specify the default load out on the ordnance tab, I believe. Thus, I was unable to test missile combat but was able to fire at Mercury with some of your lasers. NOte that your cruisers start out orbiting Earth, not MErcury, so it takes about 6 hours for them to get into beam range.
Hope this helps a little, though combat is still an area that could use some accessibility support. The actual controls are fine, it's just a little hard to tell where things are at the moment.
Also, don't forget to turn on your active sensors before you fire at anything.