FTL and there's just absolutely NO fucking way to win this game. Seriously: none. I have tried over and over - even on Easy mode - and there's just no way to beat.
I give up.
It's possible to beat (though I've only done it once or twice total on normal, and my success rate on easy is still under 50%)... it relies on a combination of luck and grinding. If the stores don't show up with the gear you need, you are hosed, of course. If the random events happen that you don't have the crew/equipment to do the blue options, you're often boned as well. And there's not much you can do to affect the happenstance of those two factors.
However the one thing you can do is grind, grind, grind.
Try to explore each sector as much as you can and time it so that you FTL jump out of the map barely 1 step ahead of the rebel fleet (or even if you have to fight the fleet at the exit it's not the end of the world so long as your ship is pretty buff). Try to keep your ship repaired up to 2/3rds hull but don't constantly boost it to full - random events that repair your hull happen pretty often and you don't want to waste scrap on repairs that could go to upgrades when you are likely to get at least a couple free repairs per sector. Until you've beaten the game at least once, I wouldn't worry too much about drones or beams (unless you happen to get lucky and get a drone control for free, in which case, try to find/prioritize defense drones). In combat, try not to use autofire if your weapons don't have the exact same charge time. Wait until they're all charged, then pause and do a massive barrage at once to blow through the enemy's shields and do the most hull damage you can - and prioritize enemy weapons control as your main target, following up with shields once that's down, and diverting a little fire to their engines only when they're trying to escape. Don't bother with bomb weapons or beam weapons - multifire lasers are your best friend, and you don't really need more than 1 torpedo launcher (and if you have to choose between mounting a torpedo launcher and a laser that shoots 2 or more times per volley, pick the laser). Of course, I'm assuming you're manning the mannable stations - having crew at weapons and shields makes all the difference especially later once they level up a couple times, and at engines helps too. If you get boarded, send everybody but your pilot into med bay and open all the doors except the medbay and cockpit to vacuum. Chances are the boarders will scurry into the medbay too, but a fight there favors you because your guys will constantly heal during the fight. Once they're in there you can hit "close all doors" so the ship starts to reoxygenate. Fight any slaver you encounter but always accept his surrender - free crew is good. Try to get 1 rock and 1 engi crewmember. The difference in repairing with an engi is huge, and between the two of them they open up the most blue options in random encounters.
By the time you get to fight the final boss, you will absolutely need full power shields, at least 6 shots per weapon volley (but more is better), and if you can lay hands on a cloaking device or defense drones, so much the better. Prioritize the enemy's torp launcher (I think it's third weapons cluster from the left), but once you've killed off the crew in the compartment (need upgraded sensors to see when the crewman dies) and broken it, it won't repair during that round any more and you can move on to other targets. Try to save your cloak for the "energy surge detected" warning periods. Heavily upgraded and manned engines can save your life here, as they will help you dodge both lasers and torps - but be careful not to go too much into diminishing returns on evade% per level bought.
Also, the first time I managed to beat the game was using the engi ship.