I know it's not as good but does that mean you should leave it to the AI? Somehow I don't think so...what is the most effective way of 'microing' battles?
Personally i prefer focus firing on the cap ships first. Once the shield is gone the mitigation is gone and the enemy ship takes full amount of damage

I was fighting an early vassari fleet with LRMS and a kol and kept on losing (around even numbers) even when I had repair platforms up so I was wondering what kind of coordination do I need.
Ive fought odds 2 to 1 against me at one time and won. It really depends on how you use what you have at your disposal. It could also mean that the enemy had better technology then you, so not much you can do but micro the fleet. Keep an eye on the empire tree and watch the health of all your ships. If it looks like your lossing health way too fast compared to your enemy, it would be better to retreat.
When you retreat and he is attacking a planet of yours, be sure to have some gun platforms as he moves after your retreating ships the gun platforms will still be firing on his ships.
So either he will chase your ships and wont be focus firing on the platforms and they will do lots of damage to his ships, while he does minimal damage to them.
Or he could focus fire on the platforms and your fleet can retreat safely and resupply at the near by planet.