From Halopedia, the Halo wiki
This Is Your 7 AM Wake-Up Call
Once you exit the cryo tube and retrieve Cortana, you should retrieve a pistol you can find on the floor (to the left of where you begin, near Cortana's holotank). Proceed through the level to the elevator. Open it, and enter the shaft. Hold up left stick to move upward, and when the Chief glances to the side, jink the left stick in that direction. Proceed upward while dodging debris to get up the shaft, tapping the left stick in the direction you wish to go. At the top, you are engaged by an Elite with an energy sword. Follow the button commands (melee) and hurl him down the elevator shaft. Continue until you are outside the observation deck. You will see an Elite several yards directly ahead on a slightly raised platform at a console with his back to you. Charge him, get an assassination or beat down, and use your pistol or assault rifle to finish off the Grunts below you on either side of the room. Cortana will tell you to hit a button on the console that the Elite was fiddling with, but first, stock up on ammunition. There are several ammo crates in the room, mostly close below you with several off to the side. Fill up your pistol, stock up on grenades, and swap your Assault Rifle for a Plasma Pistol. After you've prepared yourself, hit the button on the console. Soon after, two Phantoms will flank you and place two boarding craft on the windows. Frag the areas where the covenant jump out, kill the grunts with headshots, and EMP the Elites' shields with your Plasma Pistol before killing them with your Magnum as well. After clearing the area, follow Cortana's waypoint to the elevator. There will be several small engagements with groups of Covenant, each basicaly just an Elite and two Grunts. Play it safe, but there really isn't much to worry about. Just remember to swap out your Plamsa Pistol when you get the chance. Once you get through (I'm gonna say three?) of these groups, you'll pass an ammo crate with pistols and assault rifles. Grab ammo, and go into the next room. There will be an elite and three grunts. Follow standard tactics again, and once you've killed them, two doors will open at the back of the room, and a Jackal will come out of each. Once again, standard tactics: you can shoot their hands, make them flinch, and then kill them with a headshot; or you can grenade them, or simply punch them to death. After this engagement, I'd recommend going back to that ammo crate to fill up your pistol, and swap out your Plasma Pistol. Proceed on, through the doors the jackals came through. You'll be in a circular room, or maybe more like a hallway, that surrounds another circular room. If that's confusing, then I'll say it's basically a donut. Inside the "hole" there will be a group of grunts. head in and take them out first. They're your easiest targets. flanking the room are two jackals, one on either side. Again, standard tactics. Just make sure they don't kill you with a supercombine, as at least one has a needler. After killing both, exit the hole (if you haven't already) and continue on to the far side of the room. You will have to kill three Elites, one at a time as they come at you. The first is a Minor with a storm rifle. The second is a major with a storm rifle. The third is a major with a concussion rifle. n00b combo them all. After killing the third one, enter the passage the Elites came through, and the airlock will be to your left. Before entering it though, some Battle Rifles will appear on the wall. Swap your Magnum for one, then enter the airlock, and wait for the depressurization.
In Space, No One Can Hear You Gurgle Upon Death
Once you exit the airlock, immediately swap a Battle Rifle for your Magnum. Start sniping any Covenant you see, but be wary of the Beam Rifle-toting Jackals. Also, if the reticule turns red, and you don't think you'll kill them outright, DO NOT TAKE THE SHOT. It's better to get a guaranteed headshot than waste 3-5 rounds shooting them in the torso. Also, a forewarning, the Needler can now supercombine on YOU, so if you see anyone with it, kill them. Don't worry about the cruiser reaching you before you fire the missile, you have all the time in the world. As you clear out enemies, move around the wrecked walkway when it is safe to do so. It may be tempting to jump in and kill any remaining enemies, but you'll just end up signing your death warrant. Be careful of the Elites, though. Be prepared with plasma grenades, and a charged Plasma Pistol (grab a replacement from any dead Grunt or Jackal should you run low), should you need them. As you progress around the perimeter of the ship, always be on the lookout for any remaining enemies. When you get up to the missile controls, you'll see a Jackal and an Elite. Try and stick a grenade onto the Elite. Failing that, use a Plasma Pistol to drain his shields, then nail him in the face with a Battle Rifle round. As for the Jackal, just shoot him in the face. If your Battle Rifle is low on ammunition, swap it for the Jackal's Carbine Now, activate the missile controls. Immediately after doing so, SPRINT towards the missile that's emerging. If you manage not to get caught in a missile battery or a Covenant Supply Depot, you should reach the blast door before the Covenant's boots even touch the ship, which will set off a cutscene.
Abandon Ship! Close Down The Circus! Evacuate The Zoo!
Turn around and run. The Gravity Well won't affect you, specifically, but look where you walk, because it affects the scenery, and you don't want the last thing you see to be a Covenant antennae array punching through your chest. Get to the airlock on your right. As you enter, wait for the room to pressurize. Then run like crazy through the halls, but being mindful of the explosions. Just because Cortana rewrote the suit's firmware doesn't mean you're immune to explosions. Don't bother killing any Covenant, the ship will do that for you. Just keep sprinting towards any waypoints you see. Once you see the Elite limping down the hallway, you've almost made it. Turn left to reach a sealed door. Mind the Grunts don't shoot you. Then open the door and watch as the final cutscene plays.