@SexyToad: Everyone else has already given you good tips, in terms of A) try to zig zag around each sector, only leaving the sector when the Rebel fleet is JUST about to reach the exit beacon, and B) play on easy while you're still learning, and C) try to conserve resources like missiles whenever you can get away with it.
The other tip I'll give because some new players have trouble fending off boarders: use your door system. They gave it to you for a reason. You can manually click on each door to open them, or you can click the buttons in the lower right of the screen to open/close all doors. If you're up against some really tough boarders, open your airlocks and the doors on most of your ship, forcing the oxygen out of those areas, causing the enemies to avoid those rooms because the vacuum is deadly. Using that method, try to force them to fight you in your medbay, where you have a distinct advantage. It's a pretty cheap strategy, but this game tries to rob you of victory at every turn, so fair's fair.
Also of note: fires need oxygen to burn. If there's a fire on your ship, it's often easier to open the airlocks and a path of doors to that fire, flushing the oxygen (and the fire) out into space. Only put fires out with your crew when you absolutely have to.
Aside from that, just mess around with different upgrades, different equipment, different strategies (as far as what enemy systems to target, what things to buy first), and just spend some time getting a feel for the game. Also, if you see a blue option, always take it, because it means you're getting a more advantageous option based on a certain crew member/upgrade/item your ship has.
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