Targetting

After destroying a target, weapons will go inactive, and you have to wait for a while before reactivating them (delay depends on the firing rate). They don't switch to a new target automatically (drones might, but rather randomly :).

The only way to control what targets your weapons are firing at, is to manually change targets (by clicking on the targetted icon on the top of your screen) and activate the weapons you want to fire at this target. In the main, it's best to focus fire on a single ship.

There is no way to make the weapons automatically change target and start shooting, this is all manual control. As for the delay, this is determined by the Rate of Fire of the weapon: a gun does instant damage and no matter if the gun fires on the same target or another, the fastest it can shoot is determined by its Rate of Fire. So no, there is no way to quickly shift the fire from a destroyed target to a new.

You should be targeting as many enemies as you can so that you always have one targeted when the last one is destroyed. The ship will make the "next" target active so that you can just activate your weapons (F1, F2, ...).

The text above is credited to community members from Eve forums here.