The Earliest Dungeons and Adventures

The first ever dungeon and arguably the birthplace of role playing games was Dave Arneson's Castle Blackmoor Dungeons, before there were D&D rules, played with a proto-ruleset based on the Chaimail fantasy battle tabletop rules. This was later published by Judges' Guild as "The First Fantasy Campaign".

From experiencing this, Gary Gygax then created the dungeon under Greyhawk Castle for his own group, and after playing it intensively, wrote and published D&D. 

The first play session in Greyhawk Castle was described by Gygax: It was in the late fall of 1972 when I completed a map of some castle ruins, noted ways down to the dungeon level (singular), and invited my 11-year-old son Ernie and nine-year-old daughter Elise to create characters and adventure.[6] 

The monsters first encountered, by son Ernie's and daughter Elise's characters, were a nest of scorpions in some rubble in the very first room of the dungeon they entered. The glint of coins was mentioned to lure the incautious hand into attack proximity, but Elise's PC used a dagger to poke around, and the scorpions were spotted. Eventually one managed to sting, but the poison saving throw was made. They next encountered and defeated a gang of kobolds with a chest of 3,000 copper pieces. Needless to say, they weren't pleased with the treasure. [6 #1836] 

Gygax also recounted the scorpions as giant centipedes, which seems more probable looking at the OD&D level 1 encounter table that had those, but no scorpions: Gygax citation: 

[The first monster ever killed was] "a giant centipede, with the 1st level PCs played by my son Ernie (fighter) and daughter Elise (cleric). #4335

I recall the adventure fairly well as it was my first as a DM. Later in the long session of exploration, the two intrepid adventurers came upon the lair of several kobolds, slew two and the rest fled. They found an iron chest filled with coins...several thousand copper pieces--that was too heavy to move. A big disappointment, that, for the centipedes had been nesting in a pile of refuse in which there was located a nice piece of jewelry. #4329

Ernie's character was Tenser. After they went upstairs I stayed in my study and went to work on a second dungeon level. The next night, Don Kaye and Rob and Terry Kuntz joined in, rolling up the character Murlynd, Robilar and Terik respectively."

The second play session

Early evening and the first adventurers to explore the ruins of Greyhawk castle set forth. All were humans. Rob and Terry were fighters, Robilar and Terik, Don and Ernie were the magic-users Murlynd and Tenser, Elisé was a cleric. [...] The group slew a few rats, found stairs leading down, and entered the dungeon with torches flaring. What was that noise? (Early on I used a lot of illusionary sounds in the dungeons [...]). Anyways, all were tense, edgy and wide eyed. No surprises, a pit avoided, much exploring and mapping, and -- amongst other denizens of the dark underground trashed -- a group of kobolds decimated and sent fleeing, and a great treasure chest full of copper pieces hauled out. Great disappointment that, for they had deemed it the best of the loot, and 'twas but a trifle.[16]

Blackmoor was later published by Judge's Guild as "The First Fantasy Campaign", and one can track many of the rules in OD&D back to this material. Some of the player maps by Dave Megarry from back in the day also still survive. But Blackmoor Castle never ascended to the same fame as  Greyhawk Castle. Maybe, because it was published. Not only did this kill speculation, the real issue was that there was not much written detail on the dungeon, so the whole thing looked bland and disappointing. Below are example room descriptions from Blackmoor: 

• 7) 32 Kobolds: AC 7, 1 HTK [hits to kill, i.e. HD]
• 5) 6 Jewels, Sword +1, Armor and Shield +1, Scroll: 1 - Detect Magic, 2 - Sleep
• 9D) 2400 GP, 40 Goblins: AC 6, 1 HTK

We know that the level of notes for Greyhawk Castle was similarly sparse, but it was never published. 

The first published adventures

D&D supplement 2 Blackmoor was published in 1975 and contained the first published adventure module of all time, Temple of the Frog, which had surprisingly detailed backgrounds and descriptions. The first stand-alone adventure module was 1976's  Palace of the Vampire Queen, with one page of backstory five levels of maps with minimal room keys, and pretty haphazard encounters. This was nothing you could not just make up in an afternoon with graph paper and rolling some random encounters as suggested in the D&D rules. This is likely why it took TSR some time to understand there was a market for published modules. The first module from the Greyhawk campaign was based on a map from Rob Kuntz and distributed at a convention and later was reworked and  re-published as Lost Caverns of Tsojcanth. The published standalone TSR module which set the standard for descriptions was Steading of the Hill Giant Chief, in 1978.

The first introductory adventures

The first adventures included as examples with the game rules deserve special recognition as they define a formative shared experience for players new to the game. The original D&D rules from 1974 included a "sample level" that really was just a few rooms and corridors crammed with various traps and tricks to showcase those, not a dungeon to be explored. 

The 1977 Holmes Edition of D&D game included a dungeon commonly called the Tower of Zenopus (dissected in great detail at Zenopus Archives). The boxed set from 1978 also included a separate introductory adventure book, B1 In Search of the Unknown, which required the GM to distribute monsters and treasure, and was replaced by Gygax's classic B2 Keep on the Borderlands in 1979. Gygax did not name the NPCs in B2 and left some areas to flesh out for the GM, but otherwise it was a fully defined adventure, beloved by a generation of players. 

In the 1981 Moldvay/Cook Edition of the game, the Haunted Keep of the Rodemus family replaced Zenopus, but kept B2.  The Extended Rules came with X1 Isle of Dread, a hexcrawl adventure that could provide for months of play. 

The 1983 Metzner edition of basic D&D (the famous "Red Box") had a small 8-room solo adventure dungeon, and an introductory Group Adventure commonly referred to as "Get Bargle", wich had hook of a 1,000 gp reward for the capture of Bargle, the renegade magic-user. It consisted of a keyed ruined castle level, a map and monster suggestions for a second level dungeon, and suggested key features for a third level including Bargle who only was described as chaotic, level 5-7. It much later was fully fleshed out in Dungeon #150 as Kill Bargle and led to a host of cameo appearances for Bargle.

The 1e AD&D Dungeon Masters Guide contained an example dungeon map, only called "A sample dungeon" and after a symbol key and random encounter tables for two different areas of the map, indicating where the lairs of these factions were, under "Monastery Cellars & Secret Crypts" detail descriptions for the first three rooms, and a play narration for them too. Like "Get Bargle", there are later independent efforts to flesh out the rest of the map, but interestingly, some fantastic sleuth work demonstrates that this sample dungeon was based on the same source material as the Moathouse Dungeon from T1, Village of Hommlet.

There is a nice blog that discusses all the early TSR publications in chronological order, and here are comparisons of original D&D with basic D&D and the differences between editions of basic D&D

[References: see Greyhawk References]