Final Fantasy III/The Airship Nautilus

Let's summarize the equipment you found for free in the last few quests:
 * Full equipment for Dragoon (two different spellcasting spears, armor, helm);
 * Full equipment for Viking (spellcasting hammer, armor, helm);
 * Full equipment for Geomancer (status-inflicting bell, armor, power bracers);
 * Some rare arrows for the Ranger.

The next dungeon is miniaturized, so physical attacks will be useless. Not only that, status-inflicting weapons (Earth bell, Medusa arrows) are uneffective, too. You definitely need as much magic as you can.

Keep one Dragoon equipped with spellcasting Wind & Thunder Spears but no shield, keep the Scholar (DS) or the Ranger (NES) for healing, get the Geomancer for his Terrain ability. Finally, head to Replito and purchase the five summon spells for the Evoker: even the Chocobo Escape will be useful in a miniaturized dungeon. For the rest, the Evoker equips the same rods and armor parts as the Black Mage.

Replito and Duster
Before leaving Saronia onboard the Nautilus, purchase some equipment for the Evoker: one Light Rod for weapon, then both the Black robe and the White robe. These two robes improve the wearer's intelligence or spirit, respectively, therefore they slightly unbalance the outcome of the Evoker's Summons towards one end.

Hop onto the nautilus and head to Replito. This town is north-west of Saronia. There you can purchase five Summon spells for the Evoker. There are some free items hidden here and there.

South-East of Saronia is the island of Duster. In this town you can purchase two feather hats for the Geomancer and the Evoker. There are two Earthen Drums to be found in the grass. There is no need to purchase any other weapon or armor now: better ones will be found for free very soon. Later, you will come back here to purchase a Bard vest.

Doga's manor
Fly south from Duster and get to the Dalg continent. On its south-west end is the entrance to the windy canyon that leads to Doga's manor. In the manor you can buy Level 6 magic. From the manor the next dungeon can be accessed.

Cave of the Magic Circle
A miniaturized dungeon. Enter, explore one floor to find the shortest route through it, then exit, rest and re-enter to explore deeper. Some enemies are weak against Wind, so the Dragoon's wind spear will be useful. Luckily, there is no boss at the end. The Magic Circle at the end allows to enchant the airship Nautilus for underwater travel.

Doga's village
Optional area. You can buy Level 7 magic spells here. Access from underwater on the east side of Dorga's continent.

Temple of Time
Underwater dungeon. It is located south-west of Saronia. It conceals the Magic Lute necessary to wake up Unei. There are many locked doors and also a good knife for the Thief, so you may consider temporarily switching one character to this class. There's no boss at the end.

Sunken Cave
Optional underwater dungeon. It is located under an island east of Goldor's manor. It contains several useful pieces of equipment, including strong weapons for the Dragoon and the Viking.

Saronia Catacombs
Optional underwater dungeon. It is located, quite predictably, in the sea near Saronia. Defeat the boss, Odin, to obtain a Summon spell.