Also, if you're playing with a joystick and haven't done this for a while (or an age *g*), two tipps:
a) Give it time. After a while - and this may take several missions (you might want to replay one over and over in the simulator) - you will get better.
b) Experiment with the two joystick-settings in the options - sensitivity and deadzone. Even one point more or less can make a big difference. If the joystick is set up too sensitive then you will overshoot and have a hard time flying in a straight line, or lining a straight shot up. If's it too unsusceptible your turning rate will be too slow and you will have to move the stick more in order to aim and fly.
Also, you have to decide how to fight. Full guns is a good idea for really hurting an enemy, but your fire-rate will be slower and unless you did not assign more power to the weapons you will run out of weapon energy after jsut a few shots.
With your current trouble you might want to activate only one gun. The pirates in the second mission are quite weak, so constant fire with just the standard laser will do. No need for the ion pulse cannon.
Later, when you get better and your hit-ratio goes up, you can make your shots count, doing single salvos with full guns for more damage.
Best way imho: Assign a joystick button to select your primary weapon. When your energy drops after the first few shots simply switch from full guns to one gun and keep firing.