The Floating Continent is home to more than humans. Various creatures also villages or towns that the Warriors of the Light visit over the course of the first half of their journey. For simplicity's sake, this page will cover all of them
Tozus is the land of gnomes, a race of miniature people. The village is so small, in fact, that the Warriors of the Light (+ Desch) must use the Mini spell in order to enter it. This is the first instance of the game mandating that the player use the Mini spell and could be considered a warm-up for later instances - one of which being shortly afterward. This is further emphasized by the game providing a Flame Staff, which can be used as a weapon for free casts of the Fire spell in the village
While the village itself holds no significance, it is the source of Gnomish Bread: A type of bread that has the same ability as the Sight spell. The first volume of the NTT Final Fantasy III Perfect Guide suggests that the gnomish bread was created out of necessity - as gnomes are tiny, maps become a very neessary item.
The village's other primary point of interest is a tunnel leading through the Miranos Range to the Miralka Basin, the home of the Vikings' Cove. Dr. Shelco, the village's only doctor, has the passage within his house.
The Living Woods are home to Faries as well as trees with which they communicate. The focal point of the Living Woods is the Elder Tree - who gives to the woods and who, by the time the Warriors of the Light reach the forest, has been cursed by Hein and turned into his fortress. Upon Hein's defeat, the Elder Tree returns to the woods and announces his intention to rest for 1000 years. During that time, none can enter the forest. This isn't just flavor text either: Once the Warriors of Light leave, they cannot go back to the forest.
As the description implies, the Living Woods isn't so much a 'town' or a 'village'. It's a single-screen area that is largely optional to visit. Still, the idea of a forest with sentient trees would be revisited to some degree in Final Fantasy V with the Forest of Moore.
Gulgan Gulch is a cove hidden away from the rest of the Floating Continent. It is home to the titular Gulgan, seers who lack eyes and have willingly exhiled themselves. The Gulgans have a somewhat notable role as they're the ones who state the prophecy mentioned in the game's intro. Despite this, visiting them is optional, although useful for getting a copy of the Toad spell if the player doesn't have one already.
Much like the Living Woods, Gulgan Gulch isn't so much of a 'village' as a gathering.
The Dwarven Hollows are a cavernous town located on an island on the outer sea of the Floating Continent. As the name suggests, the dwarves reside here. The Cave is not only contains a lake leading to a Subterranean Lake, but is also located in proximity to the Molten Cave (previously the Fire Crystal's tower).
The main attraction of the Dwarven Hollows are the Horns of Ice. Gutsco the thief has his eye on the horns and by the time the Warriors of Light arrive, he's stolen one and fled to the lake. The Warriors of the Light retrieve it and return - but it turns out to have been a ploy. Disguising himself as a shadow, Gutsco tails the Warriors and takes advantage of the Dwarves removing the barrier around the remaining horn to abscond with both, before fleeing to the Molten Cave. After he is defeated once more, both horns are returned to the Dwarves.
Like Tozus, the Dwarven Hollows is an actual town, with an inn and item shops. The dwarves also have a treasury filled with oodles of items, that the warriors gain access to after returning the horns.