Well I looked up real quick the quest and it says you can finish each of the elite dungeons in any mode. From what I understand story mode isn't too bad so you should be able to do it all on story mode. The only elite dungeon that I know how to do is ED2. I hardly remember anything about ED3 (other than I was good at killing the second boss there) and just blew through ED1 in story mode with some others even though I didn't know how to actually do it haha. In any case just do story mode and be safe.
However I can give you tips about ED2 since I have done this dungeon before. Some basic items you'll need is prayer pots or super restore, weapon posion ++ (a couple of them), and antifire. Basic minimum items I would say. You need more but if it's story mode I wouldn't worry about it. When you first enter there will be a ton of jellies that will do massive damage if you let them. Protect from melee and devotion when you can. Get in a corner and get them bunched up (some people like to go into the open doorway that's nearby). Kill them with aoe abilities like corruption blast/shot and tsunami etc. After you clear them face the dragons. Make sure to stay to the right when you're facing them or you will draw the posionous spiders nearby. Keep your protect magic on and truck through them. When you clear them you come across a bridge.
These golems are like the jellies. Gather them up protect from melee and murder them all (don't forget to devotion.) Once you clear them there will be flying books and celestial dragons. protect from magic and kill the books first. After killing them kill the dragons. They have a mechanic of their own which is they heal themselves rewinding time. Just stun them with any stun ability to stop them. After you kill the several celestial dragons in your way the final one will be waiting in the boss room near the chest. He's probably the hardest boss in this dungeon. Don't worry you got this.
Make sure you have everything topped off and have a good aura on like penance or dark magicks or like manicial aura on. When you first fight him just keep your protect magic on and dps until you see a prompt it will be in text. A pulsar will show up and attack it immediately! If you don't it will proceed to hurt you badly. After killing that a glowing dark ball of death will materialize somewhere near you. Avoid it at all costs but try to get it to hit a darkhole the dragon will place (he will do this shortly before the ball of death to try to kill you. Obviously get away from the darkhole and lead the ball of death into it). After the ball of death hits the darkhole it becomes a lighthole. Get inside it since it's the only way to damage the boss at this stage. Eventually he will rain crystal rain down on you that will kill you rapidly. Just step to the other side of the lighthole to avoid this.
After this the cycle repeats and he summons the same pulsar that you need to kill right away. Just truck through and when you kill him you're done. If you need to stock up run back to chest and teleport back to 1 go up the lift and restock at the banker then go back in when you're ready. When you're ready go to chest by the lift when you entered (where the jellies were) and right click and teleport to the most bottom number. You will repeat this any time you need to bank and want to return where you left off.
After this you just face the last celestial drags..kill them like before..just keep killing these guys with your protect magic on and your antifire obviously. After going through some of them you will come across a opening to your right where a bunch of spiders are. Kill them then kill the lava strykewyrms protecting the barrier nearby (don't worry they are not nearly as tough as their wildy counterparts yet count towards slayer tasks for them). After passing through them you'll come around a corner and face black drags. Same as before just keep your protect magic and antifire up and you'll be ok. No mechanics on them they're just normal black drags that just happen to do more damage.
After wrapping them up you'll see some sort of archway nearby. Don't go on it yet it's the next boss. Make sure you have everything you need then enter( like a shield swap). This second boss loves to use some mechanics but don't worry they're not that hard. when you first enter protect magic make sure you got everything topped off and a shield swap if you didn't already. When you enter he'll hit you with a really painful magic attack; that's what the protect magic is for. Soon after you'll get a prompt saying he's smacking the ground..thats when you shield swap and resonance ASAP!! You'll resoance the damage and kill the spike that shows up next to you ASAP!
After killing the spike he'll go back to attacking you as before until he'll stop and reguriate eggs into the air. There are two types of egg stacks nearby a green pile and a blue pile. Whatever egg lands near you the color of it you go to the opposite. So for example if a blue egg(ice) lands near you, take it to the green egg pile (posion) and vice versa the other way around. Just click the pile with the opposite egg liek the example stated earlier and you'll get a prompt saying you safely did it. After this he will summon either a dragon from above that will strike you where you are with a line of fire unless you surge out of it when it happens or he'll summon a huge flameball towards you. RUN away in either case! the huge flameball becomes a flaming tornado swarm that will like devestate your health if you are anywhere in it.
This cycle will basically continue until you kill him. Once your done continue on your mission. Right click chest and click most bottom number. You'll come across a fine looking ground and see new dragon enemies in your path continuing forward. These guys have their own mechanics but they are not hard. Keep your protect magic up and antifire and start fighting them. Pay attention to the ground in front of them you'll eventually see what looks like spikes heading towards you..just step over to the side a couple spaces so you dodge it. You could resonace this hit as well but it takes adrenaline from you I think if you were to do this. Anyways just watch out for the spikes and step to the side whenever you see them about to hit you.
After killing these drags you'll come across some more and more until you evnetually reach the final boss. There will be a chest hugging the wall near the final two drags you just killed. Top off everything at the bank and keep your shield swap liek before. Now you are almost done! Don't worry this next guy isn't so bad. Make sure you have a good aura and everything topped off then when you're ready, enter.
Protect magic and approach the dragon and attack him. As soon as you do he will chase you; don't run away too much or he will melee more and it will hurt. Don't also stay right next to him..keep him at covid-19 distance and you'll be okay so just out of his melee range basically. He'll eventually jump into the air and do a tail spin it's a ranged attack so protect range briefly then switch back to magic. After this he'll either do the same thing again or say your name or something about shadows and basically do a dragon magic attack. Run away from it obviously. Periodically his magic attack will look like a black cloud swarming its way towards you. Just resonace when you can and devote when you can you'll be alright. After this for a while he'll try to spike you like the dragons earlier did. Just dodge it by steppign a couple spaces to either your left or right.
After you getting his health down for a while he'll stop and sit in the center; he's summoning four hands from the ground. Surge away and prepare to take a hit or two or more if you were close to him. Just devote when this happens. Now the hands have their own mechanic they will periodically hit you with a black magic attack similar to the third bosses but it will hurt substanitlly unless you either resonace or devote the attack. Keep alternating between both abilities and you'll be okay. In between these attacks you can soul split as well but switch on your protect magic as soon as you see the black stuff. Also when they hit you with the black magic ability step to the side because these death hands will come fromthe groun and smack you; THEY WILL KILL YOU if you don't dodge a couple steps to the left or right. At this point though you'll be able to dodge you've already face several dragons who did that to you it shouldn't be a problem.
Oh yeah make sure you are away from th eother hands and that only one is attacking you. If more than two are hitting you that spells certain death..just stay near only one at a time. Eventually after you kill all the hands the boss will come back and summon a flaming circle of death. When this happens just stay near him and soul split or just dps at this time. The circle will go around just hte one time and it will go around again but it will get closer and closer until it hist you. IF IT DOES YOU BASICALLY DIE so as soon as the second circle is starting to get uncomfortably close just escape out of there or surge away and you'll be okay. Eventually after doing this and his other attacks at like 250k health he'll fly up into the sky and say something 6 times. Each of these times stay the heck away from him. He'll spout out fire everywhere near him and this will ALSO KILL YOU QUICKLY avoid it at all costs!
at the sixth time he will touch down and flame circle again. He'll keep doing this and his other attacks form the beginning until you kill him. He won't summon the hands again so don't worry about that. He also won't fly up into the air again so it shouldn't be hard to finish him off at this point. After kill him well congrats you did it! ED2 finished! LOL. In all seriousness though if you need help with that one I can help. I probably could help with the first and third as well if we were in story mode. Good luck!