Truth to be told, the dungeons are not quite required for the Main Quest aside from the first 3 dungeons, which serve as a tutorial on dungeons and gimmicks.
However, dungeons need to be unlocked through side-quests (which are ultra-short, it's normally an NPC in somewhere in town near wher eyou turn in a main quest that tells you "dungeon over there, go check it" you go over there and unlock the dungeon). Some of those (example: Halatali) are main-quest locked, you can't open them before certain points in the story.
Others (like Aurum Vale and Qarn), however, are level-locked and you need a minimum level before the quest for them appears. These quests tend to appear in towns, though.

You're still L14, or something around that level, by level 20 the next dungeon, Halatali, should open up after the main quest even in that level range (when you learn to convert items into materia)