To the OP's core question: I suggest that you start by queuing for a specific easy dungeon, or a selection of a few easy dungeons, to get your feet wet. If you like how that goes, then you can try random dungeons instead. If you have a character under Level 50, that helps, because a group with such a character in it won't be placed in a DLC dungeon.


The easiest way for a noob to queue for and successfully complete a dungeon is to do as many of the following as you can (not all of these are possible at low levels):

• Queue as a healer.
• Wear the best magicka sets you can. Five piece sets (one for sure, two if you have them) could be selected from: Julianos, Seducer, Mother's Sorrow, Necropotence (only for pet sorcerers and wardens).
• Use one destruction staff and one restoration staff. The destruction staff should not be frost.
• Wear 7 pieces of light armor.
• Use an appropriate mundus stone (max magicka, magicka regeneration, crit chance, or -- if all your attribute points are in magicka -- perhaps max health.
• Use the best food you can get.
• Have most or all of your attribute points in magicka. Max health over 20K is wasteful even for noobs. So is any investment in max stamina.
• On the destruction staff bar slot Elemental Drain and Elemental Blockade. Cast Elemental Blockade on cooldown except in dire emergencies. Cast Elemental Drain on cooldown in boss fights.
• On the restoration staff bar slot a Regeneration morph, preferably Mutagen. Refresh this (twice to cover all four group members) on cooldown.
• Slot your class' sustain buffs -- Netch for wardens, Channeled Focus for templars, etc. Use them on cooldown.
  
  Further, for emergencies:
  • On the restoration staff bar slot a Steadfast Ward morph. If it's not Ward Ally, also slot another shield for yourself such as Dampen Magicka.
  • If a templar, slot Honor the Dead (or the other morph). If a sorcerer, slot Twilight Matriarch. If a nightblade, slot Healthy offering. If none of those, slot Healing Springs.
  • Slot a healing ultimate (resto staff or warden Trees or nightblade Soul Siphon).
  
  If you use Ward at all, use it before the others, so that it will have max size.
  
  You can do better than the above, but the above is already plenty good enough.

For this year you get 3 tickets for completing your first dungeon of the day. Does not have to be a random. For the boxes I have no idea lol

Don't even sweat it, just go along for the ride.

Groups are going to be powering through low level speed runs so fast they won't even bother taking the time to kick anyone.

If you have lower level characters, queue up with them, they'll be automatically restricted to lower level dungeons by default.

One the other hand, if it's your intent to listen to every dialogue option the NPC has to say, this will not be the time to undertake that.

I think you can try the public dungeon mob stacks solo. It could be some kind of trainee for you. At least you'll got some skills in moving and damage inside medium size stack there.
Really dungeon stacks have more damage and health, but you'll be with a group there, so don't be afraid.

As for going to the dungeon during the event. I think people will choose the simplest normals to burn it fast or the undaunted pledges to make more profit at the same time. So the fastest way to get your tickets is choosing one of these dungeons.
1. The simplest are:
Fungal Grotto I, Banished Cells I, Spindle Clutch I
There is a huge chance that there will be some high-CP player in group which will just carry you through.
2. The pledges are (we're not looking to the DLC anyway):
Nov 29: Volenfell, Elden Hollow II - bad choice for beginner, it's better to avoid the pledge today. Choose one of previously named instead.
Nov 30: Blessed Crucible, Wayrest Sewers I - choose the Wayrest Sewers I. It is simple enough.
Dec 01: Direfrost Keep, Spindle Clutch II - try to avoid the pledge today.
Dec 02: Banished Cells I, Vault of Madness - Banished Cells I is nice choice today.
Dec 03: Fungal Grotto II, Crypt of Hearts II - it's better to avoid pledge and choose a simple dungeon today
Dec 04: City of Ash I, Spindle Clutch I - Spindle is nice and simple. But you can try City of Ash I too - it is more complex but fun enough for beginner.
Dec 05: Darkshade Caverns II, Tempest Island - Tempest is more complex than the simplest ones, but it is good to try it for beginner.

I wish you luck on your starting dungeoneer way. May the damage, tank and heal be with you

https://www.elderscrollsonline.com/en-us/news/post/55271

if you want indrik tickets - any dungeon completion is fine. if you want the mysterious reward boxes - first random dungeon of the day per character comes with one. las year in several of my guilds we used to pair up super low level alts with max level characters so that we could get easy fast dungeons every single time. as lvl 10 you would only get starter dungeons. and.. completion happens on last boss kill, so skipping bosses is fine

drawback. you can just about forget getting dungeon quests completed during this even, unless you are in a premade that's willing to wait for you to turn in the quest. this is generally a problem with ESO, but it will be far more exacerbated during undaunted event, because speed - is key thanks to that ridiculous 20 hour cooldown, instead of random daily dungeons resetting like normal dailies at set time each day /grumble.

I've managed to solo (the only one I've tried to solo) Vet City of Ash I many times (was farming BSW) and I'd imagine Normal City would be really easy to do.
Vaults of Madness is another easy one.