From Halopedia, the Halo wiki
Part 01: Flawless Cowboy
And I alone have escaped to tell you
You will start inside the lifeboat that you were in the cutscene. All of the Marines in your lifeboat are dead. Walk outside and grab all the ammo you can. You will see a chasm with a river running at the bottom. There is a bridge over it and you should cross here quickly. There will be a Spirit dropping Covenant right next to your lifeboat so you should probably leave. You may stay to kill the Covenant if you wish, but they are one group you may ignore. Once you have crossed the bridge, turn to the right. You may see Banshees flying around and by sniping with the pistol and using the MA5B when it gets closer you can bring one down. If you bring one down, the other will flee. Swap your AR for the Needler.
Hopefully more Rorke's Drift than Alamo
When you get up the hill you will see a ring of rocks. Inside this ring is a party of Grunts and an Elite. Kill them however you see fit. As a reminder though, the pistol is an excellent sniper weapon against the Grunts, and even the Elite if you are careful. However, the Needler is much better. Once they are dead, proceed across the open space to a short drop and keep going. Around a turn to the left you will see a group of Marines including Sergeant Johnson, fighting a group of Grunts and an Elite. Help the Marines to eliminate the covenant forces. Sergeant Johnson will talk to you for a moment and you will be alerted by another Marine that a Spirit full of Covenant reinforcements are dropping in on the other side of the facility. This will happen two to four more times. The Elites will just look at the position where you have fired the last time, so use the rocks to reposition yourself and try to stick the Elites.
Each time, you need to kill every Covenant soldier before the next group will come. Also, you need to keep at least a couple of your Marines alive. When the last group is eliminated, Echo 419 will call in on your radio and give you a Warthog in addition to picking up the remaining Marine forces and evacuating them. Grab the Warthog and drive away from the facility. You will see a deep chasm with a ramp leading into it. There will be a tunnel and you should proceed along this until until you reach a large room. Position you Warthog at the center of the room, then use the pistol against the Grunts then Sniper Rifle for the others. In it will be an Elite, several Grunts and Jackals. On the right hand side of the room there will be a small hallway-like ramp leading upward. There will be four Elites. Jump out from the 'hog then stick the Elites. Collect plasma grenades. At the end, there will be a computer terminal. Activate the Light-bridge and head back to the Warthog. Proceed across the bridge and through the tunnels again until you reach daylight again.
Part 02: Reunion Tour
Drive up the hill and when you reach a safe place to cross, drive through the stream to the lifeboat sitting there. Next to it will be a S2-AM Sniper Rifle. You may wish to get rid of one of your weapons to grab this. The rest of the level is basically just eliminating the Covenant in each area and preserving the Marines. Echo 419 will come and pick up each group. If you get lost trying to find a group of Marines, a directional icon will appear on your HUD to help you. When you have saved your last group of Marines, climb into the Pelican and congratulate yourself.