Your landing area is no worry, as the debris from the ship you were running from wipes them out immediately. But before you leave the crash site so quickly, there's an option you need to know. To your right, on a ledge above the ship's debris, is a combat knife stabbed into the snow. If you shoot or melee it, it will light on fire and change many of the weapon caches from Forerunner weapons to human weapons. There are seven hidden knives in total, and shooting them all unlocks a hidden boss fight. If you want greater weapon selection, you can shoot the knife for more. But if you'd rather complete the level the canonical way, then ignore it and move on to the first area. Refer to here to read the location of all the knives.
The Crawler Assault
Ahead of you is a small valley with a few boulders for cover and several raised platforms. In there are a few Grunts, two Jackals with shields, an Elite Storm, and numerous Crawlers attacking them. The first area is all about headshots. Use your pistol for this area and conserve your battle rifle for now.
As soon as you enter, you'll see a Jackal fighting a Crawler. Headshot him before he turns to face you, then take out any Grunts and Crawlers near him. You'll mostly be limited to engaging from the right side of the boulder marking the "entrance", but you can climb up on top of it for some high ground. A second Jackal will be on a raised platform ahead of you; headshot him before he turns to face you. The only major danger you really have to look out for here is a Spirit dropship, which will fire some shots from its turret before flying away. If you see big blue projectiles heading towards you, fall back and wait for the dropship to leave.
It should be safe now to move up. A few more Grunts and possibly the Elite may be alive past the platforms, but they'll be distracted by the Crawlers. Take them out before headshotting the Crawlers. Be careful not too move up far, as the Crawlers' tracking shots can easily overwhelm you. Most will charge but one Crawler likes to hold position and snipe from the far end of the valley. Take it out if you see it being an annoyance.
Vale might say at this point that the area is clear, but it really isn't. More Crawlers will teleport in as you head up to the far end. They come from around a tall rock, so surprise them with a grenade and pick off the rest. After them, the area should be clear. There will be a brief Soldier encounter in the climb to the next area, but they don't fight, just flee. Before you leave, grab a new weapon to replace your likely near empty pistol. The Open Hand is a good replacement; it's hidden on a ledge to the left of where the Soldiers are seen. Climb up to the ledge and crouch walk over to nab it.
The Curving Valley
Here's where things get tough. The new Promethean enemies, Soldiers, are going to be all over this area, and most of them are going to be armed with lightrifles. They tend to fire them in bursts of two, and on Legendary can easily knock you down with just four shots. Plus, they can teleport a short distance, though thankfully it's easier to track where they're going now. They're still fairly weak in this first level, and you'll have access to the Lightrifle and thus a reliable mid-range to long-range weapon. If a Soldier flashes yellow and their face turns orange, that means their shields are down and you can deliver a headshot. Using the squad commands can make finding them easier. Your teammates may have trouble against a healthy Soldier, but can easily finish off an unshielded one.
As soon as you climb up the hill and break through the ice wall, engage the Soldier on the other side. Two bursts from the Open Hand will strip his shields, then follow up with the lightrifle for a headshot. Then before you round the corner to the valley, look off the edge to your left. You should see a piece of Mission Intel and a carbine. Arm one of your teammates with it so they can a quick-shooting long range weapon.
As you round the corner, you'll immediately be greeted by two Soldiers with lightrifles and two Crawlers. Moving up will cause more to spawn and them to fall back, so you need to take them out now so they don't overwhelm you later. To take out the first Soldier, hug the right wall for cover and use the Open Hand-battle rifle combo on him. Then for the next one move to the left side to aim at him. Take out the Crawlers when you see them to make your aim easier.
As you move up and more enemies arrive, stick to the right side to take out Crawlers. Four more Soldiers will be on the outer edges of the valley, two on the left and two on the right. They each have lightrifles, so carefully use the Open Hand-battle rifle combo on each. If you run low on Open Hand ammo, offer it to a teammate and grab another boltshot; it'll only take an extra burst to drop shields. If you run out of battle rifle ammo, use the lightrifle instead. Clear out the right side first; its greater height means the Soldiers on it will be able to snipe you from farther in the map. Then clear out the left side. From a mid to long range distance, as close range will make them either leap away or toss grenades.
When they're cleared, that leaves one last part of the valley, a hill where four Soldiers are defending the entrance to the next area. The first two Prometheans tend to be more aggressive, so they can be lured out of cover. Either break the ice walls to the right of the valley or circle around the left platform to get in easier range for your Boltshot. The last two should then be flanked via the right ice wall path. While the Soldiers are distracted by your teammates, surprise them with an attack from the side. Sometimes you can even get an assassination on them this way. When Locke says the area is clear, then you've completed the valley.
Doing this area with the knives easter egg activated doesn't change the combat much, except that now you can refill your battle rifle ammo from the dispensers around. It's a little weaker than a scoped-in lightrifle, but you should still be able to get through fine with the precision weapon-boltshot combo. Before you leave the area, pick up a Sniper Rifle hidden here. It's in a dispenser at the ridge on the right, near its highest point.
The Steep Hill, Part 1
The next two rooms are transition areas, an introduction to the Artemis and a peek at the Kraken. But before you go, there is a weapon worth noting. On the edge of the cliff where the Kraken is spotted, two special scattershots, the Loathsome Thing, can be found. One is on the left and the other is on the right. It's a fast and powerful close-range weapon, especially when scoped in, so consider grabbing it and giving the other to a teammate. If you need to trade it to do so, there are weapons by the door where you entered the cave from the security station.
As you leave the cave, you'll encounter a lance of Covenant with their backs to you, fighting a few Soldiers and Crawlers. There's an achievement for sparing the Elite, but it's not a good idea to try it on Legendary as he tends to toss grenades. If you still have a plasma grenade, stick him with it before he turns around and your team should mop up the rest. Don't try to rush them for assassinations, as oftentimes they turn around or the Soldiers they're fighting toss a grenade down.
Speaking of Soldiers, there will be three visible from where you entered. The first two are on ridges to your left and right and will likely be weakened by the Covenant a bit so they're not hard to finish off. The real challenge is a Soldier carrying a Splinter Turret, on a platform high and far from you. He needs to be taken out asap because of his explosive projectiles. If you lit the hidden knives, then use your sniper rifle to get him. If not, then use a lightrifle and keep strafing before his blasts hit you.
From here, there will be a few more Soldiers and some Crawlers. With the previous group downed, you have a lot more room to engage them, and there are numerous paths blocked by ice walls that you can break to flank them from. On the left there's a cliff you can climb to gain high ground on them, and on the right there's a ridge you can travel behind to close range. Either use your lightrifle from a distance or your Loathsome Thing up close; it's very effective on both Crawlers and Soldiers when scoped in. Using the Smart Scope while jumping is helpful here, as it lets you briefly float to shoot at foes behind cover. Only other note is to keep moving, as the Kraken in the distance will fire at you from time to time.
The Steep Hill, Part 2
When the Kraken falls back, the first half of the hill is mostly cleared. The second half is far more tricky. The hill curves to the left to reveal several Soldiers defending the temple entrance at the top. Most of them have lightrifles, and there are two on splinter turrets on the left and right. Naturally, the turrets are first priority. From cover, clear out any Crawlers that might interrupt your sniping of the turrets. Don't stay put for long or else the turrets will home in on you. Also keep your fireteam from running off in the open ground; they often do so to engage more aggressive Soldiers. Keep them back and lure those Soldiers into a trap.
When it's just the turrets and gate defenders left, use your long-range weapons to dislodge the turrets first. The spare turret left from the Soldier earlier can be used to dislodge the first turret on the right. The left one is more tricky. Use your lightrifle or sniper rifle to shoot him off. Damaging the turrets themselves may cost a shot, but it's worth it to keep others from manning them. If you activated the knives easter egg, you can use your snipes to finish off the other soldiers, since you'll find two more sniper rifles at the hill. One is on top of the cliff to the left and the other is at the edge on the right where you'd normally find a scattershot. If you didn't light the knives and don't trust your light rifle shots, then you can cross the area when the turrets are down. Run past the rocks to the ridge where the right turret was, and order your fireteam to come with you. You can climb up behind the ridge so as to engage the gate defenders. Grenades are most helpful here, and you can duck and back off to avoid their shots. Order your fireteam to push up so as to distract and force the Soldiers back.
When they're all dead, seek out some new weapons. You'll find a binary rifle in a dispenser on the right ridge behind the splinter turret. With the lit knives it'll be a railgun, and you can find another on the lower half of the hill at a left dispenser. Take it or give it to a teammate, and then take a splinter turret for yourself. It should have thirty shots in it, plenty for the final battle.
One last note is that the Kraken falling back is your only checkpoint in this area. If you clear the entire hill, don't revert checkpoint should you need to regain a small bit of ammo or such, because you'll just end up having to redo half the battle again.
The Temple Interior
The final encounter pits you against a returning foe, Promethean Knights. While they've been made tougher than they were in Halo 4, they are actually easier in a few ways. Knights can no longer teleport, don't have shields, and these are limited to the short-ranged Scattershot. They're damaged by shooting the glowing yellow sigils on their sides, then killed by shooting their exposed back weakpoint or their exposed faces when they roar.
The temple inside is a fairly small round room. There are two raised platforms to the side and a central platform with a plasma turret, but more important is the room's entrance where your team will come down from two ramps. These ramps provide the most cover and distance from the action, and so are the safest spot in the room. Inside will be several Knights, a bunch of Crawlers that phase in waves, and Covenant Grunts. The Grunts are irrelevant on high difficulties; the Prometheans will easily kill them all for you.
When the door opens, a Knight will have its back to you. The Covenant are scripted to shoot at it as it roars, but on high difficulties it's not uncommon that its armor stays undamaged. Use a weapon like the Suppressor or Lightrifle to break its armor and shoot its weakpoint. Then pick up your splinter turret and head inside. Stay towards the top of the ramp as the Crawlers come to you. Your teammates can easily deal with them, and your turret can blast them in one blow. The ones from the hill have 30 shots each, so you have plenty of ammo.
Move down the ramp to the ground floor with the turret, but watch for more Crawlers phasing in. The Knights will start coming for you, so use your turret to break their armor. A direct shot after they're unarmored can easily kill them. If you run out of turret ammo, then keep your distance and use boltshots and lightrifles. But, even if it's the Loathsome Thing, don't use a scattershot on them. Their scattershots can easily drop you in one shot at close range, while you'll have to keep pumping shots into them. So keep a safe distance.
When the Knights are all dead, head to the far door to initiate the cutscene and complete the level.
Secret Boss Fight
However, if you activated all the secret knives on Legendary, you can unlock a hidden boss fight. Activating the central knife on the plinth in the room will spawn in a giant Promethean Knight and the Warden Eternal. You will be given a plasma pistol the Prophets' Bane and active camouflage.
Before starting the boss fight, leave any weapons you want to use later by the plinth so you can quickly grab them. Also be sure that one or both of your Spartans are armed with a railgun. The Loathsome Thing is a good weapon to use later here, so set it down for now so it isn't replaced by your new ones. Then activate the knife and grab that scattershot while keeping the Prophet's Bane.
The Warden will stand by the entrance of the room and stay back until attacked. As for the Knight's, it's more aggressive but isn't any more dangerous than a normal Knight. Go for a spare plasma turret by the back of the room to the knife's left, where a Mission Intel is hidden. Use that turret to easily strip the Knight's armor and kill it off.
The Warden is much more tricky. His mobility is limited in this small room, but he's still formidable and not easy to kill with the weapons in here. Draw the Prophet's Bane and crouch-walk toward him so your camo doesn't flicker. Carefully walk behind him until you're at the door where you first entered this room. You should be right behind him and in good view of his weakpoint. Draw your Loathsome Thing, scope in, and keep shooting his weakpoint as much as you can. Done right and he'll be stunned long enough to take off half his health.
The Warden will start moving at this point and will go for your Spartans. Call them to you and have them enter the room where you first entered this battleroom. The Warden is too large to enter, though he can still shoot beams into it. From there use the lightrifles in this room to damage him, while your Spartans fire with their railguns. Sooner or later he should fall.
When he's beaten, head to the far door as usual to start the cutscene and end the level.