At this point I beat it pretty much every time I play on easy. I don't remember my first win. A key to winning is to prioritize taking out the triple rocket launcher and triple heavy laser. Kill the crewmember through fire, bio beam or teleporting, then destroy the weapon systems. In general, use the stage 1 "weak" form to kill off all buy 1 crewman, who will go from navigation to shields, wherever he's needed. Every new stage should begin with promptly taking out the rockets and heavy laser. Without those, the ship is basically helpless. Stage 2 drones can become annoying, but since you should never go into this fight without level 4 shields (which are easily attainable), it is unlikely that they alone will be able to do much to your hull.
I never rely on missiles for the final boss; lasers and beams are far prefferable, for faster hits, more damage through proper beam application (remember, it only needs to touch a tiny corner of a room to count as hitting it) and no ammo worries. Rather, try to go without rockets and trade the rockets away in favor of drones, fuel or scrap. Often you can get an extra level of shield (two bars) purely from selling off your rockets. Upgrading med bay and oxygen to level 2, as well as doors to 2 are often vital to keep intruders down, curb fires and make sure that an errant blast doesn't rob you of oxygen.
I find that either drones or a teleporter are essential. Not both, since you likely won't have power to fuel your shields, max weapons (always max them or have at least 1-2 extra bars, to prevent loss of firepower!), oxygen, medbay as well as both sub-systems. Drones should focus on defense - level 1 defense drones are helpful for taking out rockets (only emply 1, as their AI will make them target the same rocket, so no bonus, and forget level 2, as they will focus on lasers which your shields can easily deal with) and repair drones are useful for keeping your crew stationed, improving shield recharge and fire rate etc. Teleporting should focus on taking out the aforementioned rocket and laser systems. Beaming into the main hull is dangerous and often a waste of time; too many enemy crewmembers will swarm and kill your guys. Rather, once the key targets are taken care of, focus on thrusters, firepower etc. As for crew, ignore humans whenever you can. Rockmen and mantismen make good attack classes, engis are best kept at shields, where they can repair them quickly. Zoltans should be kept alive and in weapons/shields to ensure a constant energy stream there. Slugs are often worthless too, but may be used in several "blue" events. Keep them in navigation, or in medbay/oxygen for backup repairs in a crunch. Try to get as many crewmembers as you can, four at the minimum.
As a brief last note, always try to map out your trip through a system in advance. It's fairly easy to anticipate when and where you can go somewhere before the fleet tracks you down, and the more nodes you hit, the more scrap you get, and thus, the better weapons and systems you can grab. Always hit shops, try to save scrap as long as possible, sell off anything you don't actively use (don't keep "just in case" weapons or drones in your inventory, it's worthless). If possible, get a loadout that kills crew. You get a lot more resources that way, to the point where you might not actually be able to spend it all. Just remember(!) to keep something around for actually destroying hull. I once found myself with nothing but a fire beam, bio beam and crew teleporter against the boss where the AI had taken over, repairing the systems. It was no fun. I won, but it was no fun.
FTL is actually quite easy on easy mode. Especially when you get the slug cruiser. I think I spent about 5 hours before my first win, and I have probably beaten it at least a dozen times since then. Go get 'em, people.