Know Thy Enemy
While you may be used to being able to use any weapon on any enemy on lesser difficulties, it's important to realize that on Mythic, each foe is more susceptible to some weapons and tactics than others. Almost every weapon in the game serves a special purpose. Below is a list of enemies and the tactics & weapons I use to deal with each.
Grunts: A simple headshot with a Battle Rifle or Carbine will do nicely. Sniper weapons work too. The Magnum is another option but should really only be used when there's nothing better available. Make Grunts a priority since now they won't hesitate to grenade spam your location.
Shielded Jackals: Sticking their legs is one of the most effective methods for taking them out. Additionally, you can use a medium-ranged weapon to target the dent in their shields to your right, and when they stumble, follow up with a headshot. Tossing frag grenades is another way to get them to expose the meat by causing them to dive. Remember to save at least one for last, and use a plasma charge to disable its shield. Then when it turns to run, assassinate it for full shields.
Carbine Jackals: While not quite as dangerous as their sniper cousins, if given the chance they can drain your shields in the blink of an eye. Carbines are a bit risky to use against them, but the Battle Rifle is perfect due to its three-bullet spread.
Jackal Snipers: They don't nearly pack the same punch they did in Halo 2, though they can still kill you in one hit from great distances. At least this time they have lighted scopes mounted on their heads to let you know which direction they're facing. Their attention spans seem to be short as well. If one glances your way, step back behind an object, and in a matter of seconds they'll look elsewhere. Obviously using a sniper weapon is best against them, but even the BR can be used to headshot them from fair distances.
Drones: Your encounters with the insect side of the Covenant will be few and far between. The Brute Shot seems to be the most effective method for dealing with them, followed closely by the BR. When available, Brute Choppers seem to be the most effective method seeing as how it only takes two rounds to kill each one.
Brutes: Though they come in many flavors, taking care of business is pretty typical for all variations. Serving as quite a switch from Halo 2, Brutes now pack full-body power armor. Don't worry, though. It does not retain the ability to recharge and performance is vastly inferior to what we're used to from the Elites. Use a charged plasma shot to decimate their armor, then follow up with a BR/Carbine headshot. When doing this from greater distances, obviously a sniper weapon beats the BR.
Brute Stalkers: Only appearing in two levels of the game, the plasma charge won't home in on these sneaky aliens. Though it still works, you'll need to aim carefully and pray they don't strafe. In addition, the power drain has a profound effect on them, particularly when they're clustered together, and will save you a lot of trouble. However, it's almost always easier just to stick them with either a spike or plasma grenade, or you could just simply douse them with flames. Personally, I enjoy watching them burn.
Chieftains: Serving as the distinguished leaders of the Brutes, they are the battle-hardened toughest of the tough and come in two variants. Brute Chieftains always wield Gravity Hammers and usually carry Flares or Invincibility. If a certain number of Brutes in their pack are killed, they'll roar in anger and begin charging you, oftentimes activating their Invincibility. They can also spot you when you're cloaked by active camo. Taking out their shields with a charged plasma shot, then following up with roughly 10 BR shots to knock of their helmets, will suffice. Sniper weapons seem to be weak against them. It should also be noted that these particular Chieftains seem to have a glitch associated with them whereby after they use their Invincibility they become completely immune to all weapons except for lasers, rockets, and other heavy fire. In cases like this, run like a mo-fo! War Chieftains on the other hand tend to stay back with the pack and use detached plasma turrets and fuel rod cannons. Tanks and heavy weapons are your best bet here, but when not available, close the gap and circle-strafe them for an assassination. This can be done with active camo or by allowing them to get distracted by your allies.
Hunters: With their near-impenetrable armor and powerful assault cannons, your options when facing them become immensely limited. Tank shells and heavy weapons are most effective, and the Spartan Laser can kill one in roughly two hits, but it's almost always better to put your allies in between you and them. It's also possible to use grenades to cause them to dive off of cliffs on The Ark. Fortunately, you will only encounter Hunters four times in the entire game.
Infection Flood: Usually they travel in masses and can spawn from Flood Mushrooms, Flood Carriers, and even be spewed from Tank Flood, but all in all, they're only truly deadly when your shields are critical, in which case they'll swarm and have a Chiefy snack! They can also resurrect the bodies of fallen Combat Flood and downed Covenant or Marines. Peppering swarms with the Assault Rifle is one of the best counters and seems to be the only real purpose of that weapon. Alternatively, the Plasma Rifle also does a fine job of cleaning up.
Flood Carriers: Pop! Pop! It always cracks me up to see these pitiful little balloons waddling around then toppling over and blowing up. It's like a party every time you kill one! A popping Carrier Flood results in about a dozen Infection Flood being released, though the explosion itself can also kill you. Sticking them with plasmas, spike 'nades, or firebombs will not only kill the Carrier, but also the Infection Flood it produces.
Combat Flood: The three distinct forms you'll come across are Brute, Human, and Elite. The Brute and Human forms can be "headshotted" from a distance by aiming for the Infection Flood buried into the chest cavity. They also crumble easily with melees from just about any weapon. Elite Combat Flood also come in two forms, shielded and unshielded. Since after Floodgate it'll be nearly impossible to tell from a distance which type it is, meleeing with the Energy Sword is one of your best options. Remember to circle-strafe as you strike, and always let them swing first and backpedal. One of the better methods is to use a charged plasma shot at close range, then follow up with a melee. Note that Combat Flood resurrected by Infection Flood won't have shields. You can tell by the pulsing spores on their bodies. The most dangerous ones you'll encounter will be wielding Swords, Shotguns, Maulers, and Brute Shots. If you can, keep your distance and headshot them. If it's a shielded Elite Combat Flood, use the charged plasma/melee trick. In masses, the Gravity Hammer works wonders with the unshielded forms.
Stalkers (Pure Form): In this form, they have a slight lunge-melee attack, but apart from that are weak and are basically freebees for recharging your shields. They are quick, however, and can scale walls with ease in an attempt to transform into a far more dangerous form. The sword is the best weapon to use against them.
Tank Flood (Pure Form): These walking... well... tanks are near invincible to almost anything you can throw at them. I once watched one of these hulking masses get stuck by six plasma grenades simultaneously only to survive and slaughter its attackers. While flame-based weapons will always drop one, they seem to also have a strange weakness against the Energy Sword. 1-2 hits will bring one down. When using the Sword, remember to melee (not lunge!) then immediately backpedal to avoid the one-hit-kill swipe. Then follow up with another. They also have the ability to barf Infection Flood if your distance from them is too great. Nasty stuff!
Ranged Flood (Pure form): Argh! These menacing wall-crawlers have stepped up to become the most annoying foe in the game. It's already bad enough that they seem to have an unlimited amount of Carbine-like ammo that they can shoot at you super-fast, but whenever they feel threatened, they can also curl themselves up into an impenetrable ball to keep you from inflicting additional harm. And just when you're feeling gutsy enough to step out to chuck that spike grenade, it's already transformed into a Stalker and scurried around to yet another impossible-to-reach location. Alas, there's one in every game isn't there, Bungie? Whew. Okay, obviously complaining about it won't get us anywhere, so your best weapon here would be a direct stick from a plasma or spike grenade or setting it ablaze. You can also use Brute Shots or chained Needler explosions to knock them off walls. It won't kill them, but at least put you in a better position to stick 'em.
Sentinels: They only appear in the last mission and can usually be dealt with by the Arbiter, but just in case, plasma weapons will drain their shields rapidly, and a charged plasma pistol shot will always serve as an instant kill. Curiously, their beams seem to be able to flip airborne Warthogs, so you might want to avoid those ramps!