Trio is harder than Atziri by a big margin, and the only fight that has chaos damage. There will be a bunch of vaal-constructs that deal chaos damage between the bosses, those can rip you apart in the alluring abyss, apex of sacrifice trash is not that dangerous though.
Atziri Trio for most builds should be the following kill-order: Tittybitch, Dualstrike Dude, Cyclone Dude.
You always want to kill the titty-bitch first. She repositions herself after attacking a couple of times, if you haven't instantly blown her up make sure she walks into a corner (she always moves away from the player). Once she dies there's gonna be a bunch of physical degen on the ground, try to avoid it as much as possible. If she raises her arm, move away. That's the animation for her physical firestorm attack which is by far the deadliest in the entire encounter.
The Dualstrike Mob is the one that occassionally sets himself on fire. Unless you have decent survival against reflect, do not attack him during this window and focus on repositioning yourself instead. This reflect always deals fire damage. He always ignites on hit, so if you happen to be melee and have to facetank him, bring a chemists or experimenters ruby flask. If you're ranged have a dousing flask ready just in case.
The mob you generally want to kill last because he's easiest to deal with when empowered is the cyclone guy. Whenever he cyclones he will leave a chaos degen trail. This doesn't deal that much damage even with -60% chaos resist, but it's nigh impossible to see under the physical degen ground that spawns when you killed the other bosses. He will most often cyclone towards you, cleave, run after you, hit with normal melee attacks once or twice and if you try to get away from him, he will start this cycle again. Getting hit by any of his attacks will debuff you with a bunch of bleed-stacks. Bring 2 staunching flasks when melee, 1 is more than enough as a ranged/caster. Do not use them whenever you got hit, the bleed doesn't deal lethal damage unless you got hit multiple times and have 10+ stacks on you. The bleed is corrupted blood-style bleed and not puncture-type bleed so moving doesn't increase the damage it deals. You want to kill him last because the only difference between his empowered and non-empowered form is the amount of bleed-stacks he applies with each ability.
This fight is more about preparation then execution, as long as you have decent damage this fight shouldn't last that long. Don't panic and move out of sight of the bosses, you will want to see what they are doing. Depending on how controllable your damage is you may accidentally kill the cyclone-guy first and have the dualstrike dude last. This shit happens a lot once you get high damage since the cyclone dude has less HP and is faster than the other one, so he oftentimes takes the brunt of the damage. Just don't get hit while the dualstriker is enraged and don't hit him during that window. The empowered version of this asshole deals a lot more fire damage, and since all types of damage get reflected as fire damage you can easily kill yourself here. Kite while he's enraged, attack him while he isn't.
Atziri is just an afterthought after this fight. Don't kill yourself to reflect, dodge the flameblasts/stormcalls and you're good to go.