Well, It's been a while since I played it last time (Steam says, summer 2016), so I might mess up some details like weapon names.

The core principle is to be able to adapt to whatever the have is sending your way. For every ship/layout variant, there are several paths which you can exploit. Some ships are more heavily skewed towards certain strategies (like if you have a 4-person teleport pad, you DEFINITELY should employ a heavy boarding strategies), but you always have some options.

There 2 main aspects to FTL warfare: you should kill your enemy and you should keep yourself safe.

To kill enemies, there are 3 reliable ways that I've found: lasers, drones and boarding.

With starting cruiser, you already have Burst Laser II that does 3 shots, and it's arguably the best starting gun. What you need to win the game is to have 2-4 shots more than enemy shield level, to drop down their shields and then do damage to systems. The heaviest shield level is 4, so if you are capable to deliver an 8-volley barrage, you can reduce the final boss ship quite fast. If you do NOT have firepower to drop enemy shields, missiles/bombs really help, since they can bypass shields and damage the critical systems. So, for starting ship, all you have to do is to luck your way for some more firepower. You can also augment it with teleporting, if you happen to find a couple of mantis crew members. If you find some good beam weapon, you can use this instead after shields drop, targeting most dangerous systems (I always tried to disable cloaking / mind controlling structures first, but that depends.Sometimes you need to disable their drone control first, if they have some nasty repair drones or attacking drone)

Drones is another surefire way to kill the enemy. Augment a couple of Mark II offensive drones with ion weapons, and most enemy ships will be disabled 5 seconds after combat starts.

Third reliable way to win is boarding/teleporting. Just find a couple of mantis crew members (though rock guys are no slouch either), and drop them to the enemy to kill off their crew, just be ready to bring them back or do some micromanagement running away from enemy if your teleporter is disabled. And of course be sure to destroy their medlab facility, so wounded enemy won't be healed. This way requires the player to be most alert, cause you can lose your crew rather easily, but it's the most rewarding, cause you often get the ship intact, giving you the most scrap bonuses. Curiously, the last boss battle is split into 3 phases, and if you manage to kill all enemy crew on the first approach (rather hard, you need to disable their level3 healing facility), then all other phases are really easy: you just teleport your killer crew and disable their piloting, shields, and other systems really fast. You can also easily disable their isolated crew who run their offensive beam weaponry, and disabling those really helps.

That was about killing enemies. Now, about keeping yourself safe. One, evade chance really helps, so try to have your evade ratio as high as possible, with best possible pilot. Second, try to bring your shields to a lever where expected opposition can't drop it with one volley, this progressively needs a raise towards the later stages. Third, CLOAKING helps, if you have it. Time it for the heaviest volleys/missile barrage, like those launched by the final ship. Four, defensive drones rule! With Mark II defensive drone, you can practically not worry about incoming missiles. Though it's a bit expensive to maintain, unless you have a drone retraction arm (or how it's called, it allows a drone parts reuse, or something).

That's basically it. I never figured out good ways to fight with missiles, but I'm sure there is a strategy that works for them too. Hope that helps. :)

All in all, after figuring it all out, I don't remember losing a game ever if I managed to get past sector 1.

P.S. Bonus: if you ever run into a droneship that can't get past your shields, there is a gamey tactics to level up all your crew with zero risk: just reduce your offensive attacks so you hit their shields, and wait till your pilot, shields, drive and weapon operators level up after evading shots. Then rotate them in their stations, and continue. This can be achieved even on sector 1 easily, and having high level crew REALLY helps. Leveling up combat ability is not that easy, unfortunately, but doable, too.