Probably played full campaign on Legendary 4-5 times. It's good fun. =] The AI is especially good in Reach.
DMR is best weapon to have, instakills anything with a headshot, except hunter and vehicles. Ofc you need to remove the shields of Elites, and the helmets of Brutes.
Never spam shots, if you're killing Grunts and Jackals with a DMR bodyshot you're doing it wrong.
Get into the habit of finding DMR's and DMR ammo in the campaign levels. You can also drop yours for some covenant weapon, trade that for a Marines DMR, then pick yours up again.
If you don't have DMR, Pistol is the next best thing. A full needle rifle also kills lots of enemies (especially brutes). Don't spam needles though, enemies will dodge most of them, take it slow. Especially useful in Pillar of Autumn.
There's plenty of fights you can skip, and there's some fights you can have the rest of Noble Team do for you. For instance, the start of Winter Contingency you fight 5 Elites in a field. You can hang back and keep an eye on them, at some point Noble will deplete shields on one of them, allowing for you to pop out and headshot them from a distance. Further into the mission, you can just wait inside as Noble defends a room before they can shut the door.
Otherwise, most of the time you'll want to carry a plasma pistol as a secondary. Unless you have some power weapon that allows you to deal with elites (mostly Sniper, Rockets, Fuel rod, Plasma Launcher). Next best thing: Grenade Launcher also works great against shields (though it's harder to use). Next best thing: Focus rifle, plasma rifle, plasma repeater. These also work well against shields, though I'll almost always prefer plasma pistol.