The Altar Cave is the opening dungeon in Final Fantasy III. Located north of Ur it is, as the name of its musical track implies, the cave where the Wind Crystal rests.
A game's opening can give you an impression of the tone that it will take. The original Final Fantasy opens with the party directly outside of Cornelia, cutting directly to gameplay. Final Fantasy II opens into a battle against imprial troops which the party has no chance of winning, although some versions include a more formal opening to properly set the scene
Final Fantasy III begins with the protagonists falling down a hole into the cave. They quickly start bickering before realizing that the cave is crawling with monsters. This sets the tone for the game quite well.
As a first dungeon, the Altar Cave doesn't through anything particularly crazy at the player. There are random encounters, but if the party gets low on health, there is also a spring that will restore them to full health.
At the end of the dungeon is the crystal room, where the Onion Knights encounter the Land Turtle. Luckily, the cave also has a couple of SouthWinds that hit with a strong ice attack: The Land Turtle's weakness. Upon defeating it, they encounter the crystal which dubs them "Warriors of the Light". They then leave via the teleporter at the back.
Something that's easy to overlook is the early rooms of the Altar Cave: Since the party begins by falling down a hole, it may not occur to the player to go back in, missing out on some of the treasure that they could otherwise get. The hole itself is also still there, meaning you can go back to where the whole thing started, though there's no reason to.
The remake makes no changes to the cave layout itself, but makes two key differences. Gameplaywise, Luneth is the only character present in the opening sequence. Thus, the sequence is more dangerous, which sets the tone for the remake. Meanwhile the crystal chamber has been gussied up, making it more distinct compared to the other shrines.
The Mythril Mines are a mini-dungeon in Kazus, accessible in the town itself. The mine initially appears to be short, but a switch can be found, leading to a back area.
The Sealed Cave is the second mandatory dungeon - it is a dungeon filled with undead monsters and the place where the firey Djinn was sealed. In the 3D remake, it is the first dungeon in which you have all four party members.
The experience this dungeon provides varies depending on the version. In the 2D versions, you already have access to the job system, and thus will be expected to put them to use. In the 3D remake, you are restricted to the Freelancer class, but the available weapons and early level magic will encourage the player to assign specific roles to the party, which is a key element of how FF3's job system is designed.
Either way, the hordes of undead in the dungeon lead to some interesting interactions. White Mages can act on the offensive with cure spells, although potentially wasteful this early on. If the player visited the left tower of Castle Sasune and won the Wightslayer, their journey through the cave will be easier. In both versions, an Antarctic Wind can be found, which helps in the fight against the Djinn.
Partway through the dungeon, the party runs into Princess Sara - contrary to her father's (and, in the remake, Ingus') concerns, she is the one hunting down the Djinn rather than being captured by him. The party decides to accompany her further in. Upon confronting the Djinn, Sara finds that the powers of darkness have made him too powerful to be sealed away directly. Thus, the Warriors of the Light weaken him enough for Sara to finish the job.
In the 3D remake, the Sealed Cave marks the end of the prologue: After the Djinn is sealed away, the Wind Crystal spirits the Warriors of the Light back to the Altar Cave and blesses them with the ability to change jobs.
The Dragon's Peak is the third mandatory dungeon in the game, albeit a comparatively simple one. It is a mountain trail that the Warriors of the Light journey up upon hearing of the vagabond, Desch. At the top, they are grabbed by Bahamut and dumped into the dragon's nest, where they encounter Desch. Upon escaping from Bahamut, the five decide to travel together.
The Dragon's Peak is home to numerous bird enemies, that can be a pain to deal with. One of the biggest concerns is their ability to turn party members to stone, although they also tend to drop Gold Needles to cure the petrified. In the 3D version, however, the enemies can serve a higher purpose for parties with a thief: The bird enemies tend to have Phoenix Downs as steals, and Final Fantasy III in general is a game with otherwise finite Phoenix Downs until the Pixel Remaster.
The fight against Bahamut requires the party flee from it, as it is too powerful to fight at this point in the game, and will likely oneshot your party. After the battle, the party leaves Bahamut's Nest and cannot return to that portion of the mountain.