Haloruns record page: http://haloruns.com/records?lb=0806 (click the timestamps for video links)
Watch Video: Halo Speedrunning Tutorial - Part 7: Reunion
1st Elevator Skip
At the very start of the mission, run to the left side of the structure onto the ramp. This first jump is the most difficult part of the skip. You can either thrust into the wall or you can run into the wall and jump as you hit it. Jumping from the right side of the ramp can give you slightly more height. Hover at the peak of your jump and clamber onto the ledge (spamming or holding your crouch button helps significantly). Once you're on top of the structure, navigate to the right side of the button and jump through the wall to the right which should put you above the map, run through the geometry and ground pound back into the map.
2nd Elevator Skip
Right before you drop into the fight, you will spawn a phantom which will fly up to a certain height then strafe left and right. This skip can be done one of three ways: you can climb the rocks to the left of the phantom, and if you are fast enough, you can catch it during its first cycle. However you can also wait for the next cycle which is about 10 seconds slower. You can also jump from the right side of the phantom, however the phantom will fly away after the first cycle so you only have one shot. You could also launch into the air by jumping and thrusting into the gaps in the tridents that the covenant use for cover and then ground pounding onto the phantom. You want to aim for the cockpit of the phantom as that is the easiest part to clamber. Hovering and crouching in the air can help. Once on the phantom, jump, thrust and ground pound to the lower area. If you are hitting a kill zone you are too far right. Now run back towards the door to hit the trigger to continue the mission. Also Make sure you are holding the sword to make the jump slightly easier.
Before dropping into the warden section you can assign your team mates to any location in the caves to avoid them distracting or shooting the warden. It also helps to wait for a checkpoint before dropping in however you should get one as you approach the warden. The warden will spawn looking in a random direction however you want him to face away from you. You can increase your odds of this happening by shooting in the direction you want him to spawn. If you approach his platform too fast he will usually spawn facing you. Sword him in the back during his spawn animation to cancel it. You want to be standing on the tiny red mushroom. Assuming the animation was cancelled properly and the warden used the right attack, you should get sent into the sky. Now just ground pound onto the platform and run to the button.
After flying through the hole in the wall with the Phaeton, turn right and begin gaining altitude until you see a small blue light. You should be at the same altitude of it before you get too close. Fly around the left of it. Once you see the clouds and background disappear for a second you can turn to the left and fly towards the guardian in the distance to end the mission. Make sure to hold A and Y as you are finishing the mission to skip the cut-scene and save about 15 seconds. If it freezes you in place by the light, you are probably too far right or your altitude is not correct. And if the mission is not ending, you didn't hit the trigger.