When I didn't have xbox live, I played the Halo 2 campaign a lot: usually solo. Cairo isn't too hard on Legendary, IF you follow specific steps to defeat enemies. I once played through the mission Cairo, on Legendary, and only died once (1).
Here are some tips:
1) find the high ground
Using an elevated landscape is easier cover then fight on the same level of ground.
2) find cover
High ground is good, but cover is important, and having both are instrumental in your victory.
3) use the right weapons
My preference was a combination of dual-wield SMGs and a battle rifle.
Grunts are easily dispatched with a single burst from the BR (battle rifle). Just use some good head shots.
Elites are a little more tricky. To save on vital BR ammunition, I would suggest hitting higher level elites (white and yellow) with short controlled bursts from your dual-wield SMGs, and then when their shield is down, or extremely low, hit 'em with your BR.
Lower level elites (blue and red) can be killed with a BR or plasma grenade.
(Note: Don't let your kill escape your line of fire. If he takes cover, use a frag grenade. A good frag toss will either knock him out of cover, or kill him.
4) learn how to attack
You have the cover, and you have the weapons. Now you must learn how to use both of them.
The key is to keep moving. You have an advantage that they don't: your bullets travel instantaneously, and their plasma does not. Remember to change your pace so that your speed is not constant. The Covenant, elites especially, will track your speed and fire where they think you will be. Example: if you are standing still, the elites will fire directly at you. if you are strafing to the left, the elites will fire to your left in an attempt to hit you: they will hit you! But if you change your speed while running, or simple stop, they will miss you! Learning how to avoid being hit plays an important roll in your fighting style.
Cover is very important to remember. Even if you are good at avoiding enemy fire by persistently dodging, it is good practice to stay out of cover no more than a few seconds: 1.5 seconds is a good number for me. Within that time frame, I can easily land 2 BR bursts on a target and take cover again. Remember, the same place that you went into cover should not be the same place that you come out of cover: they will be looking there; they will be expecting you there; they will probably be shooting there, and its not a good practice to run into enemy fire; it ends up being more embarrassing then frustrating. Cover can be achieved by either crouching behind small crates or standing up behind big crates approximately your height; even a catwalk (high ground) alone can function as cover. Your out-of-cover orientation is important as well: you may either be looking over a cover, to the side of a cover, even between covers.
So remember the basics of your attack routine: fire at target, take cover, find new orientation/move to new location, repeat.
5) prioritize your targets
This step is simple to remember: grunts first, low level elites second, and high level elites last. If a unit charges you, whatever his class or species, kill him.
There are two specific reasons you want to kill the lower level guys first: one, its easier; two, the fewer the infantry on the field, the lesser the amount of fire you have to dodge.
6) tracking your target
Your going to be on the move, and so is your target, especially the elites. Once you move to a new cover, you must be able to find your target again and attack him. This process must be repeated until you are sure that your target is dead. On occasion, you may be fighting an elite, you take cover, you come out of cover, and the elite is gone: nowhere to be seen. Don't Panic: you have approximately 5 whole seconds to find him before his shield recharges. Here are some steps to follow: If your on a catwalk, look through it; sometimes they will hide below. If an elite is just out of sight, behind a crate for example, sometime you will be able to see the aura of his damaged shield, or if his shield is down you should be able to see the arcing electricity that is characteristic of a downed personnel shield. Using a frag grenade is a good way to take down enemies that are behind cover; even if you don't kill them with the initial blast, the concussive force of the blast should knock them out of their cover.
(Note: if you are attacking a target and you feel that finishing off that target would require you to put yourself in a very dangerous position, don't do it unless you feel that there is no other option. Who knows: you may get lucky. But remember, the Master Chief is on loan from Bungie, and they need him back for Halo 3 X) )
(Note: most strenuous situations can be avoided with good planning)
7) plan your path
I thought of a cool story to illustrate this point, but I'll just start explaining now. Two things to remember: plan your path between covers before you move, and learn how to move along a path while looking in another direction.
It is important to know where you are going to go before you move. Countless times I have seen players run to an area of the battle field not thinking about the consequences (myself included). Two important things to look for in an area are weapons/ammunition and cover. You don't always need ammunition if you have a good personal supply, but more often then not players die from lack of cover. Look for areas that can't be bottle-necked. If the enemy boxes you in, and you can't escape, you will have to fight them on their terms, and that's no fun. If you become trapped, you may run out of ammunition before all the enemies are killed. This will force you into a situation where you will have to fight the Covenant in hand-to-hand combat. It is possible to fight them this way, but it is not recommended. Simply avoid bad situations.
Running to new cover is good, shooting at your target is good, doing both simultaneously is even better: Follow your path and shoot your target. This takes lots of practice, because you must remember your movement path, know how to follow it without looking at your destination all the time, remember how to shoot and accurately hit your target, remember to dodge and cover, and track your target's position; but when you learn how to do all those things simultaneous, you will be like Tex from the RED vs BLUE series (without the cool simulated flips, kicks, punches, slides,... well you get the picture).
8) take advantage of special animations
Sometimes, if opportunity arises, elites will become angry and perform their roaring animation. Sometimes when you melee an elite, you will deliver a critical blow, that will stun him for a short time. These are special animation sequences that we may use to our advantage. Usually when roaring, the elites shields are up. This is a perfect opportunity to use those plasma grenades you found, because the elite can't dodge them while he is roaring. Why? You try dodging grenades while your roaring. The plasma grenade will be an instant kill, except for white elites; but three or four BR bursts should finish the job. If you deliver a critical blow, he will do the critical hit animation. Usually their shield will be down when they do this: hit 'em with a BR burst. Animations are helpful, so take advantage of them whenever you can.
(Note: Sometimes you will see elites running from place to place. This is not exactly an animation, but if that elite doesn't see you it is a perfect opportunity to stick him with a plasma grenade)
9) Don't use turrets
Turrets are not safe. Your stationary and have no cover. All you have is a big gun, stained with the blood of a thousand other players who thought that mounting the big gun was a good idea. In difficulties easy through heroic, the marines planted these turrets for the purpose of heavy support to ward off the Covenant; on legendary, the covenant boarded the station, broke into the armory, stole the turrets, and set them up: ITS A TRAP! Haven't you noticed the blood on the ground where the turrets are setup? And as far as the Covenant plasma turrets are concerned, those are the baited traps. The grunts are their bait: poor devils. Only Johnson should mount turrets.
I think that was all I had to say about that. With these tips, you should have a guide for training yourself to fight against the Covenant on the first mission. As far as other missions, fighting vehicles is iffy for me. Fighting snipers: shoot them before they shoot you, because they have like a 95% chance of hitting your head, and that is an instant kill. And as far as flood infection forms armed with sniper rifles are concerned (Mission: Quarantine Zone), that is near impossible to fight. You think Cairo is tough, flood snipers would be eternal damnation for you. I had a scorpion tank, but the terrain would not allow me to shoot the sniper before I had a chance to fight the flood anti-armor squad armed with... rockets. (- I stopped playing that mission. 'Twas terrible. Madness.
Well, I hope that was helpful, not that you were looking for help mind you.
Oh and don't worry: I've tested these strategies.
Oh, I forgot about the buggers. I usually just chuck a grenade and hid. Or just hid. X)