Stage #1 - How Not to Die Immediately
The first thing many people notice about the flagship is it's wide range of artillery, the most evil of which a triple leto launcher that has no concern for your wellbeing. The first time I met the flagship, I was too busy being pummeled by that one to actually fire off a shot.
Now, if you've just been rushing through the game, not collecting many weapons, you'll find that your weapons may not even penetrate the shields. So, to avoid this, spend as much time as you can in each sector, and make sure your ship has the ability to go past the four shields on the flagship. That being said, make sure to have four shields yourself, maybe a defense drone, and good engines.
Ion Blast Artillery
Shoots three 1-damage ion blasts. This isn't as scary as the other artillery, but don't leave it unnacounted for; if you aren't careful, it can ion two layers of shields. But if you have decent engines, it shouldn't be too much of a problem.
Shoots three 1-damage lasers. Yet again, one of the less scary pieces of artillery, but can really pack a punch if your shields are down. However, it's preferable not to destroy it, for reasons I will explain later. Just try and soak up the shots. It's worth it.
Triple Leto Artillery
Shoots 3 1-damage missiles. Ooooh boy. The most dangerous weapon in the flagship's arsenal. It would be for the best to destroy this early on, and kill the crew in there as well. Otherwise you're going to be pounded and pounded and pounded until you die.
Shoots a long-range 2 damage per room beam. I've found this artillery a bit meh. Compared to the other three, it's not that much of a danger. Of course, if the other weapons have whittled down your shields, you could be in trouble, but that's about the biggest threat this weapon poses.
Extra things to be aware of
There is also a hacking and cloaking module. Don't skimp - use a defense drone to make sure the hacking doesn't hit your weapons or shields. The cloaking is also a pain, but not that much of a problem if you can get most of their artillery down.
Not Another Gawd-Darned Auto Ship!
Don't think you can beat the flagship by killing it's crew. By all means try it if you want, but last time I checked a ship with an advanced A.I system doesn't care about crew.
In case I hadn't given enough away, the ship does come free with an advanced A.I system that enjoys repairing every little chip away you make. Sometimes this is a welcome change, but if you want something to stay down, then I'd reccomend keeping the guy in the Laser Artillery pod alive. As he/she's isolated, once the rest of the crew are dead, he can't run around making repairs.
Stage #2 - Wait, It Isn't Dead?
There's a phase TWO?!
Yup, that diabolical first phase was just the beginning. You've still got some more torture to go through before that sweet, sweet victory screen.
Ignoring that pun in the subtitle, the second phase of the rebel flagship enjoys its drones. It has a boarding drone, a defense drone, a beam drone and a combat drone. Along with this, the first of its super-weapons comes online. You will recieve an alarm saying 'power surge detected', then a maelstrom of beam and combat drones envelop your ship. Damaging the drones system will stop the main drones, but not the superweapon.
Time to board
In case you hadn't noticed, the doors subsystem kinda snapped off after the first phase. This means boarding is ten times easier. If you're playing with Crystal B or Mantis B, you can put that four-man teleporter to good use. Beam them into the room you hate most (usually the drone or shield room), then laugh.
Stage #3 - Federal Resilience
If you've gotten this far, good job. There's only 4 powerful weapons systems, a souped up zoltan shield, a crew teleporter, a powerful superweapon, a mind control system and 4 shields in the way of a Federation Victory! If you fail the rebels will rule over the galaxy with an iron fist! No pressure.
Eh, zoltan shields aren't a proble- wait...
So it's safe to say the flagship didn't skimp on it's super shield. This thing can absorb masses of damage, and can regenerate. So that means no boarding until you can get that thing down!
If you dealt with the crew in the early stages, this shouldn't be a problem. But if you weren't careful, the flagship's staff detail will pop round to say hi. While they're just humans and a basic mantis/rock combo should suffice as security detail, it can still distract your crew and be a pain.
This system is pretty annoying, and can deal some damage, but all in all isn't that dangerous. Another good thing to note is that if you have a mind control system, it can cancel out theirs. But if you don't have a mind control system and your ship is in disarray, with multiple fire, breaches and damaged systems, the rebels mind control can become a problem.
The rebel flagship has one last trick. It decides to projectile vomit lasers all over your cruiser. This is incredibly rude, and therefore they must be punished. The only real way to avoid this is to use a cloaking system, or just sit and pray your engines are good enough. But apart from that, fire at will.