This being the official starting adventure of 5th edition, it is full of tropes and classic situations from 40 years of D&D. It starts with the characters guarding a caravan from Neverwinter to Phandalin, sitting around a camp fire and introducing their characters:

The next morning Gundren and Sildar, who are the only ones with horses, decide that riding next to the ox cart is too slow. They instruct the group to deliver the supplies to Barthen's Provisions in Phandelin, where they will be paid, while they will ride ahead to take care of business. The group agrees. However as soon as Gundren is gone, they do search the ox cart just in case there is anything fishy about the transport. The supplies turn out to be genuine mining supplies. But they do find a first aid kit under the seat of the wagon, containing a healer's kit and a healing potion.

A day and a half passes in travel before the group comes upon a place where the woods press close to the trail, with a steep embankment on both side. Two dead horses block the path. Erdan vanishes into the woods, while the others, led by Krosh, approach the horses. Clearly the horses are those of Gundren and Sildar, and Krosh finds the empty leather case that contained Gundren's book. Unsurprisingly this in a ambush, and the group is attacked by 4 goblins. Unsurprisingly is the right word, as the goblins roll very low on their stealth check and don't manage to surprise anybody. Nevertheless they aren't so easy to kill, as two of them are using their special trait bonus action to hide at the end of every round, and thus fire their arrows unseen with advantage. Although not strictly necessary, Popée wants to use his Tides of Chaos power to provoke a wild magic surge by casting a level 1 spell. That kills a goblin, but gives a random result of Popée glowing in a bright light for one minute. So she takes an arrow that brings her down to 2 hit points. And as she always casts mage armor at the start of the day, she is already out of spell slots for the day.

After killing the goblins the group decides to have a short rest, on the insistence of Popée. Healing isn't plentiful in 5E, especially since Krosh refuses to learn healing spells and plays his war priest more like a barbarian. While the group rests, Erdan decides to scout ahead alone, until he nearly steps into a trap and returns to the group. Slightly restored, the group follows the goblin trail to their hideout, having found traces of a dozen goblins dragging two larger bodies. Arriving at the hideout I show off my new map reveal tool: A tablet running Autodesk Sketchbook with the map on a lower layer, and a layer painted black on top of it. While the group explores, I erase the black layer and show the map under it. Works pretty well! In front of the goblin hideout there is a cave entrance from which a small stream is flowing, and thickets to the side. The group takes care to watch and listen, and manages to hear goblins in the thicket. They approach, roll well on their stealth and manage to surprise the goblins. The goblin also roll a 1 for their initiative, so they only get to act after every player has acted twice. They never make it that far.

So the group enters the cave, Krosh ahead, followed by Landry with a torch, as the halfling is the only one not having darkvision. Shortly after the entrance to the tunnel there is a cave to the right, where Krosh finds 3 wolves chained to iron spikes. While Laurelin successfully uses animal handling to calm the wolves, Krosh hits them with his sword, while staying out of their reach. The other players join with ranged attacks. Enraged, two of the wolves manage a strength check to break their chains, but are quickly vanquished. While Landry and Laurelin use non-lethal damage to only stun the wolves, Krosh cuts their throats. You might have noticed at this point that Krosh is not a nice guy, which is deliberately how the player wants it. Behind the wolves the group finds a chimney going up, used to throw bones and rubbish down to the wolves, but they decide not to climb it.

Instead they follow the tunnel northwards until they come to a place where the main tunnel bends right, and there is a narrower tunnel with a steep rise. Taking the main tunnel Krosh notices a bridge, 20 feet up, which crosses the tunnel. As Landry is walking right behind Krosh, the goblin on the bridge easily spots the group and Krosh sees him dashing to the right. At that point the group decides to first explore the side passage, and Krosh clambers up the steep rise. However the ground isn't stable and Krosh's weight causes an earth slide. Krosh fails his dexterity save and the damage, added to previous wounds, knocks him unconscious. However as half-orc he immediately gets up to 1 hit point again, and Erdan heals him further with a healing word. But now the group hears the sound of a lot of water rushing towards them. They all manage to get to higher ground before a big wave of water rushes through the tunnel.

The group advances again along the main tunnel, where Krosh sees the goblin on the bridge again. Some group members speak goblin and hear the goblin shouting something along the lines of "the bastards are still up". They kill the goblin, but then a second wave of water rushes down the tunnel. This time Krosh is caught in the surge, failing two saves, and is flushed out of the cave. He returns and the group still presses onwards. Theren hears a goblin shouting from ahead "did we get them this time?" and shouts back in goblin "yes, okay". He succeeds his bluff check, and earns a point of inspiration for this. The group arrives in a big cave with two, now empty, pools of water and 3 goblins. They kill two of the goblins, while the third runs away.

Krosh runs after the goblin, and right into the cave of the dungeon's boss, a mean bugbear named Klarg. Klarg has a pet wolf, and besides the fleeing goblin there are two other goblins in that cave. A big fight ensues. I roll my dice in the open, and while the group accumulates a string of misses, Klarg lands a mighty blow on Laurelin, bringing her down from full health to unconscious in one blow. Krosh is surrounded by two goblins and the wolf, and also goes down. Theren administers the healing potion to Laurelin, while the rest of the group attacks first Klarg and then the others. Klarg misses his next attack, and then goes down. The group kills the goblins and wolf, and then Laurelin heals Krosh with laying on hands. The party finds the cave full of stolen supplies, carrying a blue lion logo. They also find a treasure chest with coins and a valuable jade statuette.

The group takes a short rest and then continues over the bridge into the last cave in the dungeon. There 6 goblins, one of them a leader, hold Sildar Hallwinter captive. The group quickly cuts down the lesser goblins, so the leader decides to hold a dagger to Sildar's throat and demands the group to stop their attacks and negotiate. However Laurelin apparently didn't pay attention, and just continues to attack a goblin, so the leader cuts Sildar's throat. The fight ends quickly, and having cleared the dungeon we end the session here.

In many ways the flow of this session was as intended in the 5th edition rules: 6 to 8 encounters with varying difficulty during one adventuring day, with two short rests. However I do know this dungeon very well, having watched a dozen or so different groups play through this dungeon on YouTube to get a feel for 5E. And apart from one group where a player decided to use thaumaturgy to increase the volume of his voice and shouted in the middle of the dungeon until monsters arrived from everywhere, my group was the least subtle in clearing the dungeon. They triggered all the traps, and didn't make use of any of the opportunities to advance by ways other than combat. They didn't capture and interrogate a goblin, they didn't leave the wolves alone, they didn't negotiate with the goblin leader. Apart from Theren mimicking a goblin, they always went with what we call "Plan A", meaning rushing in and killing everything. We all had a lot of fun, but there were missed opportunities to do other things than killing.