So while there are currently ten dungeons in-game, two of them are gated behind the quest chain in Suramar. I have no idea how much longer it will take to get into them, but we've got eight dungeons ranging in quality to talk about, so let's get to it.
First off, my general impression is that the dungeons in Legion are pretty fantastic. They're not terribly difficult - maybe a boss here or there that presents a significant challenge - but overall, on normal mode and somewhat on heroic they're at a pretty good newcomer-friendly quality. I have yet to run any on Mythic, but I get the impression that that is where things will get seriously tough.
We'll start with the four dungeons available at any level in the Broken Isles:
Eye of Azshara:
Honestly, this is my least favorite of the dungeons (possibly second least) because the main challenge of the dungeon is pulling as little trash as you need to. It's the sort of dungeon that can go quickly if people know what they're doing but if you have one guy who doesn't look where he's going you'll be spending a whole lot more time in there. Also, those seagulls, who aren't even hostile but have a tendency to fly into your AoE, are the worst.
I absolutely love the aesthetic they've come up with for the Nightmare, and this is a heavily-Nightmare-influenced area. There is perhaps more trash than necessary, and again you're incentivized to skip packs, but overall I find this dungeon a lot more enjoyable. It has an optional boss, which tend to frustrate me (anything that you can do in the LFG function should probably not have optional bosses - on my real-life friend's first run of DHT, the tank skipped the dragon boss. Less than 24 hours into the new expansion. Some people are the worst.) Anyway, the area around said dragon boss is similarly frustrating if you have people who don't seem to understand about not running over dragon eggs, but the dungeon overall gets a thumbs up from me.
I really appreciate that they do a very different sort of cave dungeon here. Again, there's a theme here of having skippable trash that you'll often wind up pulling accidentally (with more to come!) They manage to make this cave area feel fairly lived-in, and there are some cool fight mechanics, though some of the trash (actually the very first group with the tiny non-elites that drop poison on the ground when they die) is annoying.
Halls of Valor:
Visually stunning, Halls of Valor has a pretty epic feel to it. Though once again, we have two big areas (the fields of the eternal hunt and the central mead-hall area) where you're incentivized to avoid as much trash as you can (you can get to Fenryr's first phase with only one trash pull if you're careful.) There are two fights in the exact same place, but they use the space in different ways, which makes that ok, I think.
Moving on, we have the dungeon that unlocks in the middle of leveling up, Assault on Violet Hold.
Assault on Violet Hold:
This might be my least favorite dungeon (if it's not Eye of Azshara.) I get that by bringing back Dalaran they had to do the Violet Hold again, but I wish that they had figured out some other structure for it - maybe we'd descend deeper into the dungeons. In practice it's really just the same dungeon but with demons instead of Blue Dragons and of course new bosses.
Three of the dungeons open up at level 110.
Black Rook Hold:
Ok, now we're talking. This is one of if not my favorite Legion dungeon. It totally feels like a spooky haunted castle - even though there aren't any vampires, it has a very Dracula look to it. You begin way at the bottom in the crypts and slowly make your way up to the tallest tower. It might be a little trash-heavy, but with this and the Karazhan dungeon opening up in 7.1, I'm really getting my fix for "haunted house" dungeons (though it's a shame there was no animated armor for us to fight.)
Vault of the Wardens:
This place really feels vast. You'll actually recognize a lot of it from the Demon Hunter starting experience, including two of the bosses. The last section can be a bit frustrating, as the tank constantly gets stunned and knocked back, and Cordana Felsong is probably the most challenging of the 5-player dungeon bosses that I've faced so far.
Maw of Souls:
This one competes with Black Rook Hold for my favorite dungeon. It begins with us climbing some cliffs in Stormheim, eventually facing off against King Ymiron (who you should remember from Utgarde Pinnacle,) who seems to be feeding off the souls of the dead to give himself new life. After that point, Harbaron comes on the Nagalfar to take us to Helheim. From there, you fight your way out of the ship's brig and onto the deck, where the ship is tossing about on the waves. While I might have preferred that we see more of the sea - you really need to adjust you camera angle to see how much we're tossing about - I also know that there were complaints about the train in Grimrail Depot and people suffering motion sickness, so... oh well. Anyway, the final boss is an incredibly cool fight against Helya, who tears off the front of the ship in the middle of it.
So there we go.
I don't know how far into the Suramar quests you need to get to do Court of Stars and the Arcway (which makes it super, super annoying that engineers have to go into one of them in order to learn the max-level goggle schematics - a bit of a pattern for engineering quests being unreasonable) but I'm sure that once I have them unlocked and I actually get to run them I'll have impressions as well.
And then Karazhan, of course.