There is no established way that elves dressed.
Don't know why you feel this strongly about it, but as it happens, you are wrong, we know pletora of ways Tolkien's elves dressed.
Tolkien himself drew a picture of Beleg Strongbow from the first age, and from it we can see that the Elves of Doriath dressed in pointed shoes with what seems to be long white socks, knee length pants, and what looks like a black doublet.
In other words, the male Elves of Doriath dressed in a style clearly inspired by Tudor england, only a bit toned down, and seemingly withouth it's more outlandish/silly features(Codpieces and those frilly necks for example).
We also know that elves seems to have prefered shoes over boots, though some elves(The ones of rivendell at the very least) used boots, though wheter it was the prefered footwear even then is another question.
Many elves also were clearly described as "Robed" in the sense of wearing actual robes, though what sort of robes and how fanciful they might be is not specified.
Famously they are also known to use enchanted cloaks, which we can presume looked pretty similar to the movie version.
Overall though, Tolkien made it clear that all of his various people dressed in a variety of styles, depending on their climate and their ancestral heritage. An elf from Doriath would probably not much resemble an elf from Mirkwood for example.
As for war, there would be a bit more overlap(though some differences as i'll go over below), as all elves dressed the same way for war, given Plate armor is not a thing in the book legendarium.
Namely iron/steel hauberks of interlocked metal rings, aka Mail/Chainmail.
All elves wore these though there is some question wheter they wore a surcoat above it as well.
It is possible that the elves did wear coats of arms above their mail shirts, but there is nothing in text as far as i can tell that auggests this, and all description of family crests and symbols suggests they carried such on their shields, not their armor.
As for their helmets, we dont know the exact style, but from what i can tell, there doesnt seem to have been any mentions of a full helmets amongst them, so there seems to have been a tradition of helmets with open faces, with fully closed off helmets being more of a dwarf thing.
Finally, there is one bit of Armor speciffically noted to be unique to Earendil, so there is a question wheter this was from his elven side, or his Human heritage, so this style would either have been the style of Gondolin, or the house of hador.
That of course being what Bilbo describes as "Panoply of ancient kings".
Now a Panoply has several meanings, as it can be used to describe "a full suit of armor", but this is a later meaning, speciffically refering to a suit of full plate armor. The older meaning, and far and away the more obvious one when you take into consideration thst Earendil's son would found numenor, which was inspired by ancienct Greece and Rome, is the the breast armor worn back during antiquity.
Now, as Tolkien made clear there was never any true plate armor in the legendarium, this rules out the early breastplates from the bronze age and the greek golden age, and instead means that Earendil either wore a greek suit of scale armor around his chest(and some form of band armor over his shoulder, be it scales, or boiled leather), or Linothorax, the kind of paper armor that Alexander the great used to conquer the known world.
While the latter is possible, there is no mention of such kind of armor anywhere in the legendarium, but plenty of references to scale armor, so i find the first much more likely.
Which means that the Elves of Gondolin(or if we're being completely literal, only it's kings) may or may not have had breast armor similar to the kind of scale armor sometimes used by the hoplites of ancienct greece.