zSavage's Brief Guide on how to Kill Izaro

Shield Charge - Izaro charges at where you were when he begins his animation. Watch for the start animation, move out of the way, and punish him while he is in his recovery animation. He will not do this attack if you are close to him.

Leap Slam - Izaro winds up and jumps to where you were when he started the attack. Again, watch for the animation, move aside, and punish him while he recovers. He also doesn't use this if you are too close.

Bone Cascade - Looks kinda like glacial cascade. Often a follow up attack performed with not too much visible windup. If you are moving around this shouldn't hit you, and if it does, it probably won't kill you. If you are in melee, you should be OK.

Bony Ground - The ground erupts with bony things that whack at you and slow you down. Again, Izaro won't use this if you're close. Move out of it ASAP and watch for a big attack coming in like a charge or leap.

Green Spirit Orbs - Fired from the Goddess of Justice atop Izaro's back. Izaro has no windup for this. They are aimed at where you were a short time before they are fired. They also stay in the air for a while. Performed regardless of range. If you're melee, you should be able to tank this or flask it off. As ranged, keep moving, and you'll avoid it pretty well.

Pulse Swing - Izaro's most damaging attack and often a major source of death for my ranged char, at least. Izaro winds up, does a 270, and releases a pulse of damage from his sword. Does a lot of damage (often 2-3K in merc), which usually stuns you. If there's green balls in the air and you get hit by this (or a ball + cascade combo), it's usually end of run, unless you're very very tanky. He uses this when he has a sword.

Mace Smash: Very damaging melee attack, similar to Vaal oversoul or Voll slams. Look for the windup and get the hell outta Dodge.

These phases generally consist of some kind of summon or mechanic that you must defeat or buff Izaro in all the later fights. I'll share strategy for the ones I've seen and will update the more I find.

Before each Trial, there will be a dead end area called a Forgotten Reliquary. A trinket can be found that makes each phase easier.

The effects of the trinkets on each phase are as follows:
Charge Disruptors: The first Disruptor you use can be used a 2nd time.
Lieutenants: The first lieutenant hit is instantly killed.
Portals: The first portal hit is instantly destroyed.
Fonts: The first font hit is instantly destroyed.
Essences: The first essence hit is instantly destroyed.
Conduits: The first conduit activated doesn't reactivate.
Gargoyles: The first gargoyle hit is instantly destroyed (use on an inactive gargoyle for the lolz)
Idols: The first idol hit is instantly destroyed.

See a pattern?

Video
This is the one where everything is on the line. Failing means starting over, and victory means ascendance and a chance for a powerful enchant. Izaro will have all the buffs from his previous phases if you 'failed' them. Making him more difficult makes him drop additional keys, but you have to buff him quite a bit

Izaro will be in a slightly larger and differently shaped arena, filled with traps (any combination of two from the 6 possible ones). Much less of the floor is safe, so watch your footing.

Izaro has one new attack: When a large green circle appears around him, a corresponding circle will appear in a trap. If you're in this when it expires, you'll be teleported to the trap.

If you are melee, you can attempt to facetank this phase. Namelock the boss (highlight his name and hold down your attack button). Even if he teleports you into a trap, in most cases you'll walk right back to him immediately and keep hitting him due to namelock. Keep a finger or two over your flasks to stay alive if not super tanky. Remember, Izaro won't do a lot of his most dangerous attacks if you are close to him.

Finally, this gem:

Culling Strike is the easy MVP of the 3rd fight. Because Izaro's HP carries over between fights, this reduces the duration of the final fight by about 1/3. Socket it where you can and use it when he gets low to end things before you die.

I highly recommend having a move skill that can cross gaps.
I highly recommend binding 'Move Only' to your LMB and leaving it like that, if you don't already do this. This lets you control your character much better for the precision needed for the labyrinth.
I highly recommend instant life flasks, and a curse immune flask. If you have a flask that gives you a DPS boost to help threshold Izaro, I highly recommend that as well.
Some users have recommended using smoke mines and CWDT + Detonate Mines to teleport away when a big hit is taken.

Argus is a giant gargoyle thing that is Izaro's pet. He's somewhere in the maze, always in an arena like area behind a closed door.

Argus is pretty tanky and immune to just about every status effect. He also does a lot of physical damage.

However, Argus is a pretty darn dumb dude. He will always attack and pursue the closest thing to him, be it a golem, a totem, or another player.

If you're ranged, just let him agro your golem (or maybe a blink/mirror arrow clone for you archers). Then just shoot him.

If melee, you can use a decoy totem, allow your golem to soak some damage, or just have good armor/use immortal call.

Argus drops a Treasure Key that can be used at the end to open an additional chest.