Jumping and falling
Here are a few tips about going up or down.
Crouching to clear greater height
When trying to jump up onto a relatively high rock or something, it can help to hit the 'crouch' button as you get there. Your feet are pulled up a bit, giving you slightly better clearance. This can allow you to scale greater heights.
Just in case you didn't realize, keeping the jump button down will take you into an auto-repeat mode. It hardly feels very realistic to do, but it's there.
Drift, and softening an impact
The directional joystick actually gives you a small degree of drift control as you fall, which can help you pick your landing spot. When landing from a dangerous height or bouncing off something on the way down, you can reduce the potential damage by crouching as you hit. That makes physical sense of course; you're flexing your knees, not leaving them rigid. You can also soften a landing by trying to hit a sloping surface, such as the side of a rock. Used together, these techniques can allow you to survive some unfeasibly large falls!
The friction myth, and the true value of contact
One bit of advice you sometimes see for extreme falls is that you should try to keep in contact with the cliff or wall or whatever on the way down, to slow yourself up a bit via friction, and thus increase your chance of survival. I myself have included such advice on occasion. But while looking into descent methods off the first bridge in AOTCR, I did some simple testing and found no such slowing effect at all! I got up on the rim of the far end of the bridge and stepped off. Whether I rubbed against the cliff all the way down (pushing myself against it using the joystick) or whether I avoided cliff contact altogether, the results were the same: I reached a point just a metre or two above the ledge in about 3.8 seconds, where the game killed me. So, "friction" seems to be a myth; there's no slowing effect from glancing contact. I believe the actual reason you might want to keep in contact with a surface on the way down is purely to do with controlling the path you're taking. It's not helping you slow up, it's just helping you hit with optimal angle and position. It's easy to jump to the wrong conclusion and think your survival was because the contact slowed you up, as you'd imagine it would in reality. But if my test is representative of the situation throughout the game, it didn't!
The fall timer myth
Also in connection with falling, you sometimes see people talking about "the fall timer", as if the game is timing you from when you start going downwards and will kill you after a certain set time - unless you happen to hit something which "resets" the timer. Well, I'm pretty sure there's no such thing a fall timer in Halo. Rather, the game monitors your downwards speed, and if it reaches a certain critical value you're toast. If you hit something which reduces your downwards speed, it's not resetting a timer, it's just helping prolong the time before your acceleration under gravity makes you reach that critical 'kill-speed'. Or at any rate, that's my conception of how things are working. So I think the 'fall timer' is a myth.
Firing during a fall
When you fall, the game seems to keep a memory of the direction you were facing at the start, and it then restricts your field of fire to the forward 180° for that initial direction. If you swivel around while falling and try to shoot in a direction outside that field, the shot doesn't go there; it gets sent off to the nearest side of the aforementioned field, which can be a bit annoying! Because of this shortcoming, it can sometimes be best to start your fall by moving backwards off your starting surface, so that your field of fire will include certain enemies you want to surprise on the way down.